Unfolding CR Singularities

Adam Coffman

Author address:

Department of Mathematical Sciences, Indiana University - Purdue University Fort Wayne, 2101 E. Coliseum Blvd., Fort Wayne, IN 46805-1499

   E-mail address: CoffmanA@ipfw.edu

Contents

List of Figures vii

Unfolding CR singularities 1 1. Introduction 1 2. Topological considerations 4 2.1. A grassmannian construction 4 2.2. Global index sums 6 3. Local defining equations and transformations 9 3.1. Defining equations for m-submanifolds in ℂ^{n} 9 3.2. Parametrized families of submanifolds 10 3.3. The transformation group for unfoldings 12 4. A complexification construction 14 5. Real surfaces in ℂ^{2} 17 5.1. Normal forms 17 5.2. Unfolding CR singularities of surfaces 23 5.3. Visualization 35 6. Real m-submanifolds in ℂ^{n}, m < n 48 6.1. Normal forms 48 6.2. Unfolding CR singularities of m-submanifolds 55 6.3. The geometry of the algebraic unfolding 60 7. Rapid convergence Proof of the Main Theorem 62 7.1. A functional equation 62 7.2. A solution of a linearized equation 65 7.3. The new defining equations and some estimates 96 7.4. Composition of approximate solutions 105 8. Some other directions 110

Bibliography 113

Abstract

A notion of unfolding, or multi-parameter deformation, of CR singularities of real submanifolds in complex manifolds is proposed, along with a definition of equivalence of unfoldings under the action of a group of analytic transformations. In the case of real surfaces in complex 2-space, deformations of elliptic, hyperbolic, and parabolic points are analyzed by putting the parameter-dependent real analytic defining equations into normal forms up to some order. For some real analytic unfoldings in higher codimension, the method of rapid convergence is used to establish real algebraic normal forms.

2000 Mathematics Subject Classification. 32V40, 32S30, 58K35.

Key words and phrases. CR singularity, normal form, real submanifold.

Presented to the American Mathematical Society at the Spring 2006 Central Section Meeting.

List of Figures

1 Elliptic CR singularity 36 2 Hyperbolic CR singularity 36 3 Nondegenerate parabolic CR singularity 37 4 Deformation, t_{1} < 0, of nondegenerate parabolic CR singularity

into an elliptic/hyperbolic pair of CR singularities 38 5 Totally real deformation, t_{1} > 0, of nondegenerate parabolic

CR singularity 38 6 Degenerate parabolic CR singularity, η = −1 40 7 Degenerate parabolic CR singularity, η = +1 41 8 Deformation of the η = −1 degenerate parabolic CR singularity,

t_{1} ≠ 0, t_{2} = 0 42 9 Deformation of the η = −1 degenerate parabolic CR singularity,

t_{1} = 0, t_{2} < 0 42 10 Deformation of the η = −1 degenerate parabolic CR singularity,

t_{1} = 0, t_{2} > 0 43 11 Deformation of the η = +1 degenerate parabolic CR singularity,

t_{1} ≠ 0, t_{2} = 0 43 12 Deformation of the η = +1 degenerate parabolic CR singularity,

t_{1} = 0, t_{2} > 0 44 13 Deformation of the η = +1 degenerate parabolic CR singularity,

t_{1} = 0, t_{2} < 0 44 14 Discriminant locus in the t_{1}, t_{2} parameter space for the

unfolding ; M^{+} 46

Unfolding CR singularities

1. Introduction

The local equivalence problem for real submanifolds of complex manifolds under local biholomorphic coordinate transformations can be approached by finding normal forms for the local defining equations. We follow that approach here in the case of real surfaces in ℂ^{2} with complex tangents, and, more generally, real m-submanifolds of complex n-manifolds with m ≤ n. Then, we introduce a way to further understand the local extrinsic geometry by seeing how it changes under small perturbations of the embedding.

The general idea is to parallel the development of the singularity theory of differentiable maps, where the geometry of singularities is understood first by a classification of the local defining equations by finding normal forms, and secondly by an analysis of how the different types of singularities fit into parametrized families of maps (called “unfoldings”). The classification of unfoldings is, again, to find normal forms for the defining expressions, at least for the lowest degree terms, under an appropriate group of transformations.

Some analogies between the geometry of singular maps and the geometry of CR singularities of real submanifolds had already been noted at least since [MW] and [Webster_{1}], where a real analytic embedding of M in ℂ^{n} is related to a certain singular holomorphic map ℂ^{m} → ℂ^{n} via a complexification construction, which we recall in Section 4.

Our starting point in the analysis of the normal form problem for the defining equations of a real surface M in ℂ^{2} near a CR singularity will be the quadratic normal forms of [Bishop]. The well-known elliptic/parabolic/hyperbolic classification and some subsequent refinements are recalled in Section 5.1, and we contribute new quartic normal forms for some degenerate parabolic cases (Proposition 5.1).

Then, having some explicit equations for manifolds representing some of the simplest local normal forms, one of the basic issues in the deformation theory is the question of the topological stability of a CR singularity, that is, whether a submanifold with a CR singularity will still have a CR singularity after a small perturbation. If the singularity persists, then there is the geometric problem of what sort of properties of a given singular point will change or be preserved under perturbation.

One approach, also going back to [Bishop], uses a grassmannian variety construction to define a notion of general position, and to give an expected codimension formula for the locus of CR singularities. Section 2 will review some of the differential topology of submanifolds with CR singularities.

The bulk of this paper is devoted to consideration of the local problem, by the analysis of the local defining equations for real analytic submanifolds embedded in ℂ^{n}. To study the geometry of the deformation of a real m-submanifold M through k real parameters, we consider a real (m + k)-submanifold ; M of ℂ^{n+k} containing M as a submanifold. Briefly, if M is defined by a system of multivariable power series equations {e⃗(z⃗) = 0}, then ; M is defined by introducing new terms depending on new parameters, {e⃗(z⃗, t) = 0}. The setup of the defining equations is straightforward — the interesting part is a group action; both are described in detail in Section 3. Our classification problem is then to find normal forms for ; M under a certain group of holomorphic transformations, which respects the difference between the original (space direction) coordinates z⃗ and the new (time parameter) variables t.

Since the normal forms for n-manifolds in ℂ^{n} are qualitatively different from the normal forms for m-manifolds in ℂ^{n} with m < n, the two cases are considered separately in Sections 5 and 6, with the simplest representatives being surfaces in ℂ^{2}, and 4-manifolds in ℂ^{5} (first considered by [D_{1}], [Beloshapka], [C_{1}]).

In Section 6, we consider a non-trivial unfolding of a real analytic manifold with a certain type of isolated, degenerate CR singularity in the case \frac{2}{3}(n + 1) = m < n, and state Main Theorem 6.5, which claims the existence of a local coordinate transformation so that the unfolding ; M (and therefore also the original manifold M) is real algebraic. The Proof, in Section 7, uses the technique of rapid convergence.

As a preview of Main Theorem 6.5, we state the following special case. Let M be a real 4-manifold in ℂ^{5} given by the real analytic equations

{y_{2} = E_{2}, y_{3} = E_{3}, z_{4} = z̄_{1}^{2} + e_{4}, z_{5} = (z_{1} + x_{2} + ix_{3}^{2})z̄_{1} + e_{5}},

where E_{2}, E_{3}, e_{4}, e_{5} are higher order parts: series with terms of degree at least 4 in z_{1}, z̄_{1}, x_{2}, x_{3}. This M has an isolated complex tangent at the origin, and it will be shown in Section 6 why this is a natural set of equations to consider, and that this type of singular point is unstable under perturbation in the following interesting way. We consider a real parameter t_{1}, and a submanifold ; M of ℂ^{5} × ℝ given by

y_{2} = E_{2}(z_{1}, z̄_{1}, x_{2}, x_{3}, t_{1})
y_{3} = E_{3}(z_{1}, z̄_{1}, x_{2}, x_{3}, t_{1})
z_{4} = z̄_{1}^{2} + e_{4}(z_{1}, z̄_{1}, x_{2}, x_{3}, t_{1})
z_{5} = (z_{1} + x_{2} + it_{1} + ix_{3}^{2})z̄_{1} + e_{5}(z_{1}, z̄_{1}, x_{2}, x_{3}, t_{1}),

where a new quadratic term, it_{1}z̄_{1}, appears, and the new higher order parts are the same as the old for t_{1} = 0, but new terms depending on t_{1} have been added, in an arbitrary (but still real analytic) way. ; M will be called an unfolding of M; it has the property that ; M ∩ {t_{1} = 0} = M × {0}, and for t_{0} fixed but close to 0, ; M ∩ {t_{1} = t_{0}} is a real 4-manifold inside ℂ^{5} × {t_{0}}, which we think of as being a small deformation of M. The result of the Main Theorem is that there is a coordinate change with identity linear part, defined near the origin (0, 0):

(z̃_{1}, …, z̃_{5}) = (z_{1}, …, z_{5}) + p⃗(z_{1}, …, z_{5}, t_{1}),
t̃_{1} = t_{1} + P_{1}(t_{1}),

which is a local biholomorphic map from one {t_{1} = constant} slice to another (near the origin): ℂ^{5} × {t_{1}} → ℂ^{5} × {t̃_{1}}, and simultaneously a real analytic re-parametrization of t_{1}, so that the defining equations of the real manifold ; M in the new coordinate system are the real polynomials:

(1)  {ỹ_{2} = 0, ỹ_{3} = 0 z̃_{4} = ¯ z̃_{1}^{2}, z̃_{5} = (z̃_{1} + x̃_{2} + it̃_{1} + ix̃_{3}^{2})¯z̃_{1}}.

In this coordinate system, it will be easy to see (in Section 6.3) that slices of ; M where t̃_{1} is a small but positive constant are totally real, while slices of ; M where t̃_{1} < 0 have two nondegenerate CR singularities. Thinking of t_{1} as a time parameter, this deformation represents a pair of CR singular points moving toward the origin, meeting in a degenerate CR singular point at time 0, and then cancelling so that there are no CR singular points after time 0.

The technique for constructing such an analytic transformation, that is, finding p⃗ and P_{1} in terms of the given (E_{2}, E_{3}, e_{4}, e_{5}), is to solve a system of nonlinear functional equations, using a method of linear approximation and rapid convergence, as employed in similar problems by [Moser], [C_{4}], [C_{6}]. The exact solution of the equation is the composite of a sequence of approximate solutions, and a norm of each approximate solution in the sequence needs to be bounded in a certain way for the convergence argument to work. The approximate solutions are constructed by solving a linearized system of equations by a comparison of coefficients calculation, Theorem 7.6. This is the longest step in the proof of the Main Theorem 6.5, and the intricacy of the calculation is due in part to the large numbers of equations and variables, with the t (parameter) variables treated differently from the z, x variables. The x_{3} variable is also distinguished because of the inhomogeneity of the cubic normal form. The more subtle and more serious difficulty regards the size of the domain of the approximation. The norm of one solution in the sequence of approximations might be estimated only on a strictly smaller domain than that of the previous approximation, so this shrinking of the domain must be controlled precisely. However, the linearized system has a large solution space of formal series, which includes divergent series solutions as well as solutions that converge on arbitrarily small sets; such approximate solutions are not suitable for an iterative process with the goal of converging toward an exact solution which is analytic on an open set. The calculation of Theorem 7.6 makes some choices to avoid the bad solutions and find a good solution, defined by a series expansion which converges on a suitably large set, and with a useful bound on its norm following from estimates for subseries and applications of Cauchy’s Estimate and the Schwarz Lemma at several points of the construction.

2. Topological considerations

Mostly we will be looking at the local geometry of real submanifolds, however we start with the big picture by recalling some topological notions, without any claim of novelty with this paper, but with the intention to motivate and give a global context for some of the local constructions in the remaining Sections. These global notions will not be required in any later Proofs, but they do give a nice explanation for some of the choices of dimensions m, n, which might otherwise appear as merely technical restrictions (such as the hypothesis of Proposition 6.2) or computational conveniences (for example, Equation (105) in the middle of the lengthy calculation proving Theorem 7.6). The explicit normal forms for unfoldings give an opportunity to see, in a concrete way, how a global count of singularities can be conserved even under local deformations that create or destroy singular points, as in the example (1) from the Introduction.

2.1. A grassmannian construction.

Let M be an oriented real m-submanifold smoothly embedded in ℂ^{n}, with m ≤ n. An embedding in general position will be totally real at most points x: the tangent space T_{x}M will contain no complex lines. At other points x, the tangent space may contain at least one complex line: such a point is a CR singular point of M (a point where M has a “complex tangent” is also known as a “complex point,” “complex jump point,” or “exceptional point”). The CR singular locus is stratified by the (complex) dimension j of the largest complex space tangent to M: denote

(2)  N_{j} = {x ∈ M : dim_{ℂ}(T_{x}M ∩ iT_{x}M) ≥ j}.

Recall (see [Lai], [F], [G]) that if G is the grassmannian variety of oriented real m-subspaces in ℂ^{n} ≅ ℝ^{2n}, then the real m-subspaces T such that dim_{ℂ} T ∩ iT ≥ j form a subvariety D_{j} of real codimension 2j(n − m + j) in G. Define the Gauss map M → G : x ↦ T_{x}M, so the set N_{j} is the inverse image of D_{j}, and for M in general position, meaning that the Gauss map is transverse to each D_{j} \ D_{j+1}, the set N_{j} will have codimension 2j(n − m + j) in M. When M is in general position and its dimension is equal to the real codimension of D_{j} in G, the Gauss map will meet D_{j} \ D_{j+1} in isolated points where we can assign an oriented intersection number, giving an index +1 or −1 for each isolated point in N_{j}.

Currently, the best understood CR singularities are those in N_{1}\N_{2}, the simplest and most generic type, where the tangent space contains a complex line but no complex plane. This paper will also consider only such points, leaving the study of points in N_{2} as one of the open areas listed in Section 8. However, we will not restrict our attention to M in general position as defined above; some points x ∈ N_{1} where the Gauss map meets D_{1}\D_{2} non-transversely will be of interest. The pairs (m, n) where the dimension m of M is equal to the real codimension 2(n−m+1) of D_{1} in G satisfy m = \frac{2}{3}(n+1). The case (2, 2), where a real surface in general position in ℂ^{2} will either be totally real everywhere or will have isolated complex tangents, is considered in Section 5. When m < n, the smallest pair of dimensions where CR singularities are expected to occur is m = 4 and n = 5, where the codimension of D_{1} in G is 4, so a real 4-manifold M in ℂ^{5} in general position is totally real except at isolated points in N_{1} \N_{2}. The next cases of m = \frac{2}{3}(n+1) are (6, 8), (8, 11), …, although their local geometry is expected to behave in about the same way as the (4, 5) case since the local normal forms for nondegenerate singularities for all the pairs (m, n) with \frac{2}{3}(n + 1) ≤ m < n fall into a common algebraic pattern, as shown in [C_{6}].

When m < \frac{2}{3}(n + 1) (the case of very high codimension, such as a real surface in ℂ^{3}), CR singularities could still occur in a submanifold M, but are unstable, in the sense that one would expect most small perturbations of M to be totally real. So, we refer to the range \frac{2}{3}(n + 1) ≤ m ≤ n as the stable dimension range for N_{1} CR singularities, and the later Sections will consider only these cases.

2.2. Global index sums.

An especially interesting one-parameter deformation phenomenon to be observed in Sections 5 and 6 is a pair creation/annihilation process, where the number of isolated, nondegenerate CR singular points increases or decreases by 2. At some critical intermediate point, one expects some sort of topological degeneracy where the Gauss map meets D_{1} \ D_{2} non-transversely. This cancellation property of certain types of pairs of CR singular points has been considered by [F] in the case of surfaces in ℂ^{2}, and by [D_{1}] for 4-manifolds in ℂ^{5}, in both cases as examples of the h-principle method of Gromov: if M has a pair of CR singularities with opposite indices (and the same orientation, in the m = n = 2 case), then there is a homotopy of embeddings of M that deforms it into another submanifold with two fewer CR singular points but the same index sum. An analogue in the singularity theory of differentiable mappings would be the pairwise cancellation of cross-cap singularities of smooth maps from real surfaces to ℝ^{3} as in [Whitney_{2}]. In this paper, in Example 5.18 and Subsections 5.3, 6.2, 6.3, these cancellation processes are considered from a local point of view. At the moment of contact between colliding nondegenerate CR singularities of opposite index, there is still a CR singularity, but with a degenerate normal form.

For the ±1 intersection indices of the Gauss map as previously defined, the index sum of all the isolated, nondegenerate CR singularities of a compact oriented real submanifold embedded in ℂ^{n} is a topological invariant. More generally, the index sum can also be defined when M is smoothly immersed in an almost complex manifold, and is an invariant of the homotopy class of the immersion. Even more generally, let M be a d_{M}-dimensional connected, oriented, smooth manifold, let F be a smooth real vector bundle over M of real rank 2n, with a complex structure operator J, so (F, J) is a complex vector bundle of complex rank n, and let T be an oriented real subbundle of F of real rank m. The CR singularities are points x on the base M where T_{x} ∩ J_{x}T_{x} is a nonzero subspace (such subspaces are J-invariant, so they are complex subpaces of F_{x} with complex dimension 0 ≤ j ≤ n). It makes sense (see [DJ]) to redefine G to be the bundle over M where the fiber over x is the grassmannian of real oriented subspaces of F_{x}, to define a Gauss map M → G : x → T_{x}, and to say that T is in general position if the Gauss map transversely meets the smooth, oriented submanifolds D_{j} \ D_{j+1} of G (D_{j} defined in each fiber as above). In the case d_{M} = m = \frac{2}{3}(n + 1), M meets D_{1} \ D_{2} transversely in isolated points x ∈ N_{1} with intersection indices ind(x) = ±1. When M is compact, N_{1} is a finite set and

(3)  ∑_{x∈N_{1}} ind(x) = ∫_{M} Δ_{m}(p(T), c(F, J)),

where Δ_{m} is a polynomial in the pontrjagin classes of T, p(T) = 1 + p_{1} + … + p_{ℓ}, ℓ ≤ m/2, and the chern classes of (F, J), c(F, J) = 1+c_{1}+…+c_{n}. The polynomial expression is a Giambelli-type formula, specifically, Δ_{m} is the degree m part of the following formal quotient of total chern classes:

c(F, J) · (c(T ⊗ ℂ))^{−1}
(4)  = (∑_{a=0}^{n} c_{a}) · (∑_{b=0}^{ℓ}(−1)^{b}p_{b})^{−1}

(5)  = (∑_{a=0}^{n} c_{a}) · (∑_{i=0}^{∞}(∑_{b=1}^{ℓ}(−1)^{b+1}p_{b})^{i})
(6)  = 1 + c_{1} + (c_{2} + p_{1}) + (c_{3} + c_{1}p_{1}) +
(c_{4} + c_{2}p_{1} − p_{2} + p_{1}^{2}) +
(c_{5} + c_{3}p_{1} − c_{1}p_{2} + c_{1}p_{1}^{2}) +
(c_{6} + c_{4}p_{1} − c_{2}p_{2} + c_{2}p_{1}^{2} + p_{3} − 2p_{1}p_{2} + p_{1}^{3}) + …

Step (4) is the definition of pontrjagin classes, and (5) is a formal series expansion. The terms of (6) are grouped by total (even) degree.

In the case where the subbundle T is the tangent bundle of M, the pontrjagin classes are topological invariants of M and do not depend on its orientation, however, the integration over M to get an integer does depend on the orientation. Reversing the orientation of M will switch the indices (±1) on the LHS of (3), and reverse the sign of the characteristic number on the RHS. In the case where M is immersed in an almost complex manifold A, (F, J) is the restriction (or pullback by the immersion) of the ambient complex bundle (T A, J_{A}) to M, and the chern class c(F, J) is an invariant of the homotopy class of the immersion. When A = ℂ^{n}, the chern class is trivial, c(T ℂ^{n}) = 1 + 0 + … + 0.

Example 2.1. The case where d_{M} = m = n = 2 gives a formula of [Webster_{3}], where T is a real, oriented 2-subbundle of a complex 2-bundle (F, J) over a smooth, compact, oriented surface M. If T is in general position, the set N_{1} of points x where the fiber T_{x} is complex (or “anticomplex,” where the given orientation of T disagrees with its orientation as a complex subspace of (F, J)) is finite. The enumerative formula (3) then reads:

(7)  ∑_{x∈N_{1}} ind(x) = ∫_{M} c_{1}(F, J).

When T is the tangent bundle of M immersed in ℂ^{2}, F is trivial and the index sum is zero.

Example 2.2. In a further special case of Example 2.1, suppose M is an immersed compact surface in general position in ℝ^{3}. Considering ℝ^{3}, with real coordinates x_{1}, y_{1}, x_{2}, as a real hyperplane in ℂ^{2} with coordinates z_{1} = x_{1} + iy_{1}, z_{2} = x_{2} + iy_{2}, the only complex lines contained in ℝ^{3} are the horizontal planes parallel or equal to the z_{1}-axis. So, the complex tangents are the familiar critical points of the height function: elliptic points (local maxima and minima), and hyperbolic (saddle) points are the only types of critical points for M in general position. This coordinate system will be used in Section 5.3 to visualize CR singular surfaces in ℂ^{2}. The topological oriented intersection number counts elliptic complex points and hyperbolic anticomplex points as index +1, and hyperbolic complex points and elliptic anticomplex points as index −1.

We will recall the more general definition of elliptic and hyperbolic points for surfaces in ℂ^{2} in Section 5. The above sign convention differs from that of [F], which assigns all elliptic points index +1 and all hyperbolic points index −1 (so the only difference is for the anticomplex points). In the notation of [F], the RHS of (7) is equal to I_{+} − I_{−}. Formula (7) is not the only topological invariant of surfaces in ℂ^{2} — a formula which involves the Euler characteristic of M and also applies to non-orientable surfaces is discussed by [BF], [F], [IS] App. IV, [Slapar]. For M embedded in ℝ^{3} as in Example 2.2, the index sum using the [F] sign convention is I_{+} + I_{−} = χ(M), a familiar result from Morse Theory.

Example 2.3. The case d_{M} = m = 4, n = 5 gives a formula of [D_{1}]:

∑_{x∈N_{1}} ind(x) = ∫_{M} c_{2}(F, J) + p_{1}T.

So, for a compact, oriented 4-manifold immersed in ℂ^{5}, the index sum of its CR singularities is equal to its pontrjagin number.

For example, it is easy to embed the 4-sphere S^{4}, with p_{1}S^{4} = 0, as a totally real submanifold of ℂ^{5}, by first embedding it in a totally real ℝ^{5}. In contrast, ℂP^{2} (considered only as a smooth, oriented 4-manifold) has no totally real immersion in ℂ^{5}, since p_{1}ℂP^{2} = 3. An embedding of ℂP^{2} with exactly three CR singularities and some immersions with exactly five CR singularities were constructed in [C_{2}].

The topological cancellation of complex points of surfaces has recently found various geometric applications, for example, [DJ], [IS], [Slapar]. The relationship between pontrjagin classes and CR singularities has been considered since [Wells] and [Lai], and further formulas appear in [Webster_{2}] and [D_{2}]. For more on the interpretation of some of these characteristic numbers in terms of Giambelli-Thom-Porteous formulas for degeneracy loci of bundle maps, see [HL] and [C_{1}].

3. Local defining equations and transformations

Here we set up the general framework and notation for the local defining equations of real submanifolds, and the action of the transformation groups.

3.1. Defining equations for m-submanifolds in ℂ^{n}.

For the rest of the paper, m ≤ n and M is assumed to be a real analytic embedded m-submanifold of ℂ^{n}, which has coordinates z_{1} = x_{1} + iy_{1}, …, z_{n} = x_{n} + iy_{n}. It follows that for any point x ∈ N_{1} \ N_{2} (as in (2), where there is exactly one complex line tangent to M at x), there is some translation taking x to 0, and some complex linear transformation taking the tangent space T0M to the real subspace with coordinates x_{1}, y_{1}, x_{2}, …, x_{m−1}, which contains the z_{1}-axis. Further, in some neighborhood of 0, M can be written as a graph over its tangent space:

(8)  y_{2} = H_{2}(z_{1}, z̄_{1}, x), …,
y_{m−1} = H_{m−1}(z_{1}, z̄_{1}, x),
z_{m} = h_{m}(z_{1}, z̄_{1}, x), …,
z_{n} = h_{n}(z_{1}, z̄_{1}, x).

where the functions H_{2}, …, H_{m−1}, h_{m}, …, h_{n} are real analytic (defined by convergent power series with complex coefficients, centered at the origin) functions of x_{1}, y_{1}, x_{2}, x_{3}, …, x_{m−1}, or equivalently, z_{1}, z̄_{1} = x_{1} − iy_{1}, x = (x_{2}, …, x_{m−1}). Until Section 7, the size of the domain of convergence will not be of concern, only that the series are assumed to converge on some open neighborhood of the origin. The functions H_{2}, …, H_{m−1} are real valued (as functions of (z_{1}, z̄_{1}, x)), and the functions h_{m}, …, h_{n} are complex valued. Since M is tangent to the z_{1}, x space at 0, the series for H_{2}, …, h_{n} have no constant or linear terms, so each function could be labeled O(2) according to the following Definition.

Definition 3.1. A (formal, with complex coefficient C) monomial of the form Cz_{1}^{a}ζ_{1}^{b}x^{I} has degree a + b + I, where I = (i_{2}, …, i_{m−1}) is a multi-index, and a + b + I = a + b + i_{2} + ⋯ + i_{m−1}. A power series in m variables e(z_{1}, ζ_{1}, x) = ∑ e^{abI}z_{1}^{a}ζ_{1}^{b}x^{I}, is said to have degree d if e^{abI} = 0 for all (a, b, I) such that a + b + I < d. Sometimes a series of degree d will be abbreviated O(d).

Definition 3.2. Similarly for n variables, a monomial Cz_{1}^{a_{1}} ⋯ z_{n}^{a_{n}} has degree a_{1} + ⋯ + a_{n}, but we will also work with the “weight,” a_{1} + ⋯ + a_{m−1} + 2a_{m} + ⋯ + 2a_{n}. A series p(z⃗) = ∑ p^{a_{1}...a_{n}}z_{1}^{a_{1}} … z_{n}^{a_{n}} has “weight” W if p^{a_{1}...a_{n}} = 0 when a_{1}+⋯+a_{m−1}+2a_{m}+⋯+2a_{n} < W.

Once T_{x}M is in standard position with local defining equations (8), the normal form problem is to find representatives of equivalence classes under biholomorphic changes of coordinates. Holomorphic transformations fixing the origin and the tangency of M to the (z_{1}, x_{2}, …, x_{m−1}) subspace take vectors z⃗ = (z_{1}, …, z_{n})^{T} to

(9)  z̃ = (z̃_{1}, …, z̃_{n})^{T} = A_{n×n}z⃗ + p⃗(z⃗),

where p⃗(z⃗) is a column vector of n functions of n variables p_{1}(z⃗), …, p_{n}(z⃗), each of which is holomorphic in a neighborhood of 0 ∈ ℂ^{n} and has no constant or linear terms (so the weight is 2), and where A, the invertible linear part of the transformation, has matrix representation of the form

A_{n×n} = ( \begin{matrix}a_{1} & a_{2} … a_{m−1} & a_{m} … a_{n} \\ 0 & R_{(m−2)×(m−2)} & C_{(m−2)×(n−m+1)} \\ 0 & 0 & C_{(n−m+1)×(n−m+1)} \end{matrix} ) .

The block R has all real entries (and is invertible because A is), and the entries a_{1}, …, a_{n}, and in the blocks C are complex. The defining equations in the new z̃ coordinate system will still be of the form (8) but the goal is to find normal forms that expose the geometry of the equivalence classes.

3.2. Parametrized families of submanifolds.

The approach to constructing a deformation of a CR singular real submanifold of ℂ^{n} will be to consider it as a slice of a higher-dimensional real submanifold of a higher-dimensional complex space. The deformation is parametrized by k real parameters, labeled t = (t_{1}, …, t_{k}). For the sake of convenience in describing the ambient space, the parameters are the real parts of k new complex coordinates w_{1} = t_{1} +is_{1}, …, w_{k} = t_{k} + is_{k}, although all the action will take place in the real subspace where s_{1} = … = s_{k} = 0. Consider ℂ^{n+k} with (column vector) coordinates (z⃗, w⃗) = (z_{1}, …, z_{n}, w_{1}, …, w_{k})^{T}, and a real submanifold ; M of ℂ^{n+k} with defining equations in a polydisc centered at (0, 0) given in the form of a graph over the real (m + k)-space with coordinates z_{1}, x_{2}, …, x_{m−1}, t_{1}, …, t_{k}:

(10)  y_{2} = H_{2}(z_{1}, z̄_{1}, x, t), …,
y_{m−1} = H_{m−1}(z_{1}, z̄_{1}, x, t),
z_{m} = h_{m}(z_{1}, z̄_{1}, x, t), …,
z_{n} = h_{n}(z_{1}, z̄_{1}, x, t),
s_{1} = … = s_{k} = 0.

For a fixed vector t_{0} (sufficiently near 0 ∈ ℂ^{k}), let M_{t_{0}} = ; M ∩{w⃗ = t_{0}} be a slice of ; M. Then M_{t_{0}} is a real m-submanifold of the n-dimensional complex space {w⃗ = t_{0}}, and clearly any M ⊆ ℂ^{n} as in (8) is of the form M_{0} for some ; M as in (10). If k = 1, we can think of t = (t_{1}) as a real time parameter, so that we are considering ; M as a time evolution of M = M_{0}, and in general, we think of each M_{t_{0}} ⊆ ℂ^{n} × {t_{0}} as a submanifold “close to” M when t_{0} is close to 0. The slice M_{t_{0}} need not be CR singular, as examples will show.

We will consider only ; M which is real analytic and which contains the origin, so that the real analytic functions H_{2}, …, h_{n} have no constant terms, and then by a complex linear transformation of ℂ^{n+k} that fixes the w⃗ coordinates, we may assume that M_{0} is actually of the form (8), so that H_{2}(z_{1}, z̄_{1}, x, 0), …, h_{n}(z_{1}, z̄_{1}, x, 0) are O(2) in the z_{1}, z̄_{1}, x variables. This means the defining equations are of the form H_{2} = r_{2}^{α}t_{α} + O(2), …, h_{n} = c_{n}^{α}t_{α} + O(2), where r_{2}^{α}, …, r_{m−1}^{α} are real coefficients summed over α = 1, …, k, c_{m}^{α}, …, c_{n}^{α} are complex coefficients, and the O(2) notation now extends Definition 3.1 to include t in the count: the degree of z_{1}^{a}ζ_{1}^{b}x^{I}t^{K} is a+b+i_{2} +⋯+i_{m−1} +k_{1} +⋯+k_{k}.

A complex linear transformation of the form z̃_{1} = z_{1}, z̃_{σ} = z_{σ} − ir_{σ}^{α}w_{α} for σ = 2, …, m − 1, z̃_{u} = z_{u} − c_{u}^{α}w_{α} for u = m, …, n, w̃_{α} = w_{α} for α = 1, …, k, will transform the defining equations of ; M to eliminate the linear terms in t, so that H_{2}, …, h_{n} in (10) are all O(2). Such a transformation fixes ℂ^{n} × {0} pointwise and acts as a translation on each parallel n-space ℂ^{n}×{t_{0}}. Since M = M_{0} is not changed and each M_{t_{0}} is merely translated without changing its geometry, we will from this point work only with ; M in \underline{standard position}, where the linear coefficients r_{σ}^{α}, c_{u}^{α} have been normalized to 0, and the tangent space T_{(0,0)} ; M is the subspace with coordinates z_{1}, x_{2}, …, x_{m−1}, t_{1}, …, t_{k}.

Evidently, ; M itself is a CR singular (m + k)-submanifold of ℂ^{n+k}, and its tangent space at (0, 0) contains a complex line, but no complex plane.

3.3. The transformation group for unfoldings.

Given an unfolding ; M, one could try to put its defining equations (10) into a normal form by a biholomorphic transformation of a neighborhood of the origin in ℂ^{n+k} in analogy with (9), but instead it is proposed to work with a subgroup of the full transformation group, that, roughly, preserves the distinction between the variables parametrizing M = M_{0} and those parametrizing the deformation.

Notation 3.3. Let B_{m,n+k} denote the group of germs near (0, 0) of transformations of ℂ^{n+k} that fix the origin, are biholomorphic in (z⃗, w⃗) near (0, 0), and whose linear part preserves the tangent space T0 ; M as in (9). Let U_{m,n,k} denote the subgroup obtained by further imposing the requirement that the transformation (11) preserve (near (0,0)) the real subspace {s_{1} = ⋯ = s_{k} = 0} as a set, so that vectors of the form (z_{1}, …, z_{n}, t_{1}, …, t_{k})^{T}, with the last k entries real, are taken to vectors of the same form.

Sometimes we will abuse notation by referring to transformations instead of germs, and spaces such as ℂ^{n} instead of neighborhoods of the origin, although in Section 7 more care will be taken with the precise size of the domain of convergence of holomorphic maps.

The requirement that the subgroup U_{m,n,k} preserves the real subspace as a set means that its elements are transformations of the form:

(11)  (z̃, w̃) = A(z⃗, w⃗) + p⃗(z⃗, w⃗),

where the invertible linear part acts on column vectors

(z⃗, w⃗) = (z_{1}, …, z_{n}, w_{1}, …, w_{k})^{T}

by the (n + k) × (n + k) matrix representation

(12) A = ( \begin{matrix}a_{1} & a_{2} … a_{m−1} & a_{m} … a_{n} & a^{1} … a^{k} \\ 0 & R_{(m−2)×(m−2)} & C_{(m−2)×(n−m+1)} & R_{(m−2)×k} \\ 0 & 0 & C_{(n−m+1)×(n−m+1)} & 0 \\ 0 & 0 & 0 & R_{k×k} \end{matrix} ) .

and where the higher degree part is of the form

(p_{1}(z⃗, w⃗), …, p_{n}(z⃗, w⃗), P_{1}(w⃗), …, P_{k}(w⃗))^{T} ,

with the quantities P_{1}, …, P_{k} real valued functions on the subspace {w_{1} = t_{1}, …, w_{k} = t_{k}}. Without the requirement that the t entries remain real, the last row of the above block matrix A would be 0, R_{k×(m−2)}, C_{k×(n−m+1)}, R_{k×k}.

As a consequence of the defining property of U_{m,n,k}, the subgroup has the following property: for a fixed t_{0} ∈ ℝ^{k} (near 0) and transformation (z⃗, w⃗) ↦ (z̃, w̃) in U_{m,n,k}, all the points in the complex affine subspace {w⃗ = t_{0}} = ℂ^{n} ×{t_{0}} (near (0, t_{0})) are taken to another complex space ℂ^{n} ×{t̃_{0}}, with constant w̃ = t̃_{0}. In particular, the subspace ℂ^{n} ×{0} is fixed as a set (near (0, 0)). So, if we think of M_{t} ⊆ ℂ^{n} ×{t} as a t-dependent perturbation of M = M_{0}, then the transformations under consideration are those that, for each t are holomorphic transformations of the ambient space of M_{t}, ℂ^{n} → ℂ^{n}, and simultaneously the quantity t may be real analytically re-parametrized. The restriction of the transformation to ℂ^{n} × {0} is exactly of the form (9), so it will be convenient to arrange for normal forms of ; M that put M into already known normal forms.

In analogy with the deformation theory of singularities of maps, where the mapping variables and deformation parameters are transformed by groups that respect the distinction between the two types of coordinates by taking certain fibers to fibers ([AVGL] §§1.1, 3.1, [Lu] §3.4, [Martinet], [PS] §6.1, [Wall] §10), ; M ⊆ ℂ^{n+k} and any of its normal forms under transformations of the form (11) will be called a k-parameter “unfolding” of the CR singular submanifold M. Manifolds ; M_{1}, ; M_{2} ⊆ ℂ^{n+k} related by a transformation in the above subgroup will be called u-equivalent.

Having set up the framework for the defining equations and transformation groups, from this point the reader could skip ahead to either the treatment of real surfaces and their unfoldings in Section 5, or to the case of higher codimension in Section 6 (which is different, and treated independently, from the surface case). In the next Section, we continue a general discussion of real analytic defining equations and their complexification.

4. A complexification construction

The previous Sections mentioned some analogies between the geometry of CR singularities and the theory of singularities of maps. It remains to be seen whether the analogies are merely rough and superficial, or can be made precise, and this paper will not give a complete answer to that question, the goal being instead to consider some of the simplest examples. As one possible framework for an investigation of the correspondence between CR singularities of real submanifolds and critical points of holomorphic maps, we recall a complexification construction ([MW], [Webster_{1}]), which relates a given real m-submanifold M ⊆ ℂ^{n} to a certain parametric map ℂ^{m} → ℂ^{n} (or, more precisely, a neighborhood of the CR singularity in M to the germ of a map near the origin of ℂ^{m}). Our analysis of normal forms in later Sections will not explicitly refer to this construction, unlike [MW], where the geometry of the complexification is crucial for the theory. Instead, we will be using this complexification technique as a computational tool, for identifying the CR singular locus of some concrete examples in Subsections 5.3 and 6.3.

First, for an arbitrary power series h = ∑ h^{abI}z_{1}^{a}ζ_{1}^{b}x^{I} as in Definition 3.1, let ℏ = ∑ ℏ^{abI}z_{1}^{a}ζ_{1}^{b}x^{I} be the series with coefficients defined by the formula ℏ^{abI} = \overline{h^{baI}}. Then, the function ℏ(z_{1}, z̄_{1}, x) has the property that it is equal to \overline{h(z_{1}, z̄_{1}, x)} when ζ_{1} = z̄_{1} and x = (x_{2}, …, x_{m−1}) is real.

Next, we want to consider the equations (8) of M, which are in the form of a graph over the tangent space, in two different ways, both as a parametric map and an implicit description.

M can be described implicitly in ℂ^{n} by the following 2n − m equations:

(13)  0 = y_{2} − H_{2}(z_{1}, z̄_{1}, x), …,
0 = y_{m−1} − H_{m−1}(z_{1}, z̄_{1}, x),
0 = z_{m} − h_{m}(z_{1}, z̄_{1}, x), …,
(14)  0 = z_{n} − h_{n}(z_{1}, z̄_{1}, x),
(15)  0 = z̄_{m} − ℏ_{m}(z_{1}, z̄_{1}, x), …,
(16)  0 = z̄_{n} − ℏ_{n}(z_{1}, z̄_{1}, x).

Equations (13–14) are exactly the equations (8), of which the y_{σ} = H_{σ} equations are self-conjugate. Equations (15–16) are the complex conjugates of the z_{u} = h_{u} equations. Now, consider ℂ^{2n}, with coordinates (z⃗, ζ⃗) = (z_{1}, …, z_{n}, ζ_{1}, …, ζ_{n})^{T}. Taking Equations (13–16) and replacing every occurrence of z̄ with ζ gives the following 2n − m equations:

(17) 0 = \frac{z_{2} − ζ_{2}}{2i} − H_{2}(z_{1}, ζ_{1}, \frac{z_{2} + ζ_{2}}{2} , …, \frac{z_{m−1} + ζ_{m−1}}{2} ), …,
0 = \frac{z_{m−1} − ζ_{m−1}}{2i} − H_{m−1}(z_{1}, ζ_{1}, \frac{z_{2} + ζ_{2}}{2} , …, \frac{z_{m−1} + ζ_{m−1}}{2} ),
0 = z_{m} − h_{m}(z_{1}, ζ_{1}, \frac{z_{2} + ζ_{2}}{2} , …, \frac{z_{m−1} + ζ_{m−1}}{2} ), …,
0 = z_{n} − h_{n}(z_{1}, ζ_{1}, \frac{z_{2} + ζ_{2}}{2} , …, \frac{z_{m−1} + ζ_{m−1}}{2} ),
0 = ζ_{m} − ℏ_{m}(z_{1}, ζ_{1}, \frac{z_{2} + ζ_{2}}{2} , …, \frac{z_{m−1} + ζ_{m−1}}{2} ), …,
0 = ζ_{n} − ℏ_{n}(z_{1}, ζ_{1}, \frac{z_{2} + ζ_{2}}{2} , …, \frac{z_{m−1} + ζ_{m−1}}{2} ).

Near (0,0), these equations define a complex analytic m-submanifold M_{c} embedded in ℂ^{2n}. Denoting

Δ : ℂ^{n} → ℂ^{2n} : z⃗ ↦ (z_{1}, …, z_{n}, z̄_{1}, …, z̄_{n})^{T} ,

the set {ζ_{1} = z̄_{1}, …, ζ_{n} = z̄_{n}} = Δ(ℂ^{n}) is a totally real 2n-subspace of ℂ^{2n}, and M_{c} ∩ Δ(ℂ^{n}) = Δ(M). Denoting π : ℂ^{2n} → ℂ^{n} : (z⃗, ζ⃗) ↦ z⃗, (π ∘ Δ)(M) = π(M_{c} ∩ Δ(ℂ^{n})) = M. In a neighborhood U of 0 ∈ ℂ^{n}, π(M_{c}) is a complex analytic variety (possibly all of U) containing M.

M can also be parametrically described as the image of the parametric map σ : ℝ^{m} → ℂ^{n} :

(z_{1}, z̄_{1}, x) ↦ (z_{1}, x_{2} + iH_{2}, …, x_{m−1} + iH_{m−1}, h_{m}, …, h_{n})^{T} .

The composite Δ ∘ σ : ℝ^{m} → ℂ^{2n} is a parametric map:

(z_{1}, z̄_{1}, x) ↦ (z_{1}, x_{2} + iH_{2}, …, h_{n}, z̄_{1}, x_{2} − iH_{2}, …, \overline{h_{n}})^{T} .

The complexification of the parametrization is the holomorphic map Σ : ℂ^{m} → ℂ^{2n} defined by replacing z̄_{1} with ζ_{1} and x = (x_{2}, …, x_{m−1}) with complex variables ξ = (ξ_{2}, …, ξ_{m−1}) in the expression for Δ ∘ σ:

Σ : (z_{1}, ζ_{1}, ξ) ↦ (z_{1}, ξ_{2} + iH_{2}(z_{1}, ζ_{1}, ξ), …, h_{n}(z_{1}, ζ_{1}, ξ),
ζ, ξ_{2} − iH_{2}(z_{1}, ζ_{1}, ξ), …, ℏ_{n}(z_{1}, ζ_{1}, ξ))^{T} .

Let δ denote the inclusion of the totally real subspace

(18)  \{ζ_{1} = z̄_{1}, ξ_{2} = \overline{ξ_{2}} = x_{2}, …, ξ_{m−1} = \overline{ξ_{m−1}} = x_{m−1}\}

in ℂ^{m}; then Σ ∘ δ = Δ ∘ σ and π ∘ Σ ∘ δ = π ∘ Δ ∘ σ = σ. For a neighborhood V of the origin in ℂ^{m}, Σ is a holomorphic embedding V → ℂ^{2n}, and Σ ∘ δ is a totally real embedding. The connection between the parametric and implicit descriptions is that Σ is a local parametrization of the complex submanifold M_{c}: the image Σ(V) is equal to M_{c} near 0 ∈ ℂ^{n}. The composite π ∘ Σ : ℂ^{m} → ℂ^{n} :

(19) (z_{1}, ζ_{1}, ξ) ↦ (z_{1}, ξ_{2} + iH_{2}(z_{1}, ζ_{1}, ξ), …, h_{n}(z_{1}, ζ_{1}, ξ))^{T}

is holomorphic, but not an embedding: the complex Jacobian drops rank along a critical point locus which includes the origin. For a point u⃗ = (z_{1}, z̄_{1}, x) in the image of δ in ℂ^{m}, if u⃗ is not a critical point of π ∘ Σ, then π ∘ Σ is a local holomorphic embedding near u⃗, which takes a neighborhood of u⃗ in the totally real m-subspace (18) to a totally real neighborhood of (π ∘ Σ ∘ δ)(u⃗) = σ(u⃗) in M. So, if σ(u⃗) is a CR singular point in M, then δ(u⃗) is a critical point of π ∘ Σ.

This complexification construction has the practical advantage that the critical points of π ∘ Σ are easy to compute, and can be considered ‘candidates’ for CR singular points of a given real analytic parametrization. Generally speaking, the critical points of a complexified parametric map are also candidates for differential-topological singularities of the real map, and such points were observed in [C_{2}] and [C_{3}] for some parametrized images of real varieties in complex projective space. However, the local defining equations set up in Section 3 are always, being graphs of smooth functions, smooth embeddings.

In Sections 5 and 6, we will also complexify the defining equations (10) of ; M, treating the real t coordinates as more of the real x coordinates, so x and ξ = (ξ_{2}, …, ξ_{m−1}) in the above expressions will be extended to (x, t) and (ξ, ω) = (ξ_{2}, …, ξ_{m−1}, ω_{1}, …, ω_{k}).

The map π∘Σ is the one mentioned at the beginning of this Section; it seems from the form of (19) that there should be some connection between the geometry of its critical points, in terms of the betterunderstood singularity theory of holomorphic (or smooth) maps, and the geometry of the CR singularities of M. The classification of singularities of maps up to certain group actions may relate to the classification of CR singularities up to biholomorphic transformations (9).

However, the actions of the groups B_{m,n}, B_{m,n+k}, U_{m,n,k} (9, 11) on the defining equations of M and ; M (8, 10) are not exactly the same as the group actions usually considered in the theory of singularities or their unfoldings ([AVGL] §3.1, [Martinet]). So while there may be a useful analogy between the singularity theory of maps and the geometry of CR singularities, it is already clear from previously known examples, and those to be considered in the next Sections, that the details will not be the same, and an understanding of the group action would be the key to making the analogy precise.

5. REAL SURFACES IN ℂ^{2} 17

5. Real surfaces in ℂ^{2}

As previously mentioned, a real surface in ℂ^{2} in general position is either totally real everywhere, or has isolated CR singularities.

5.1. Normal forms.

We begin by recalling some well-known normal forms for real surfaces in ℂ^{2}. The defining equation in standard position as in (8), with the manifold tangent to the z_{1}-axis, is:

z_{2} = h_{2}(z_{1}, z̄_{1}) = Q(z_{1}, z̄_{1}) + C(z_{1}, z̄_{1}) + e(z_{1}, z̄_{1}),

where Q and C have terms of only degree 2 and 3 respectively, and the series e = O(4).

Proposition 5.1. For any real analytic CR singular surface M there is exactly one normal form {z_{2} = Q+C+e} from the following list equivalent to M under a holomorphic transformation of a neighborhood of the CR singularity.

Label  Q  C  e  comment

elliptic generic

z_{1}z̄_{1} + γ(z_{1}^{2} + z̄_{1}^{2})  0  O(4) real 0 < γ < \frac{1}{2}

elliptic cusp

z_{1}z̄_{1}  z_{1}^{3} + z̄_{1}^{3}  O(4) real  γ = 0

elliptic higher cusp

z_{1}z̄_{1} 0

\begin{matrix}z_{1}^{s} + z̄_{1}^{s} \\ +O(s + 1) \\ real \end{matrix} \begin{matrix}γ = 0 \\ s ≥ 4 \end{matrix}

elliptic circular paraboloid

z_{1}z̄_{1}  0  ≡ 0 \begin{matrix}γ = 0 \\ s = ∞ \end{matrix}

parabolic nondegen.

z_{1}z̄_{1} + \frac{1}{2}(z_{1}^{2} + z̄_{1}^{2}) i(z_{1} − z̄_{1})z_{1}z̄_{1}  O(4)  γ = \frac{1}{2}

parabolic degenerate

z_{1}z̄_{1} + \frac{1}{2}(z_{1}^{2} + z̄_{1}^{2}) 0 \begin{matrix}η(z_{1}^{2}+z̄_{1}^{2})z_{1}z̄_{1} \\ +O(5) \end{matrix} \begin{matrix}γ = \frac{1}{2} \\ η = ±1, 0 \end{matrix}

hyperbolic generic

z_{1}z̄_{1} + γ(z_{1}^{2} + z̄_{1}^{2})  0  O(4)

\begin{matrix}\frac{1}{2} < γ < 1 \\ or \\ γ > 1 \end{matrix}

hyperbolic nondegen. diophantine

z_{1}z̄_{1} + z_{1}^{2} + z̄_{1}^{2} (z_{1} − z̄_{1})z_{1}z̄_{1}  O(4)  γ = 1

hyperbolic degenerate diophantine

z_{1}z̄_{1} + z_{1}^{2} + z̄_{1}^{2}  0  O(4)  γ = 1

hyperbolic fold

z_{1}^{2} + z̄_{1}^{2}  0  O(4)  γ = ∞
cubic  0  C  O(4)

Proof. The only new bit of information in the above table is the quartic part for the degenerate parabolic case, so that is the only calculation skteched here. The computation is typical of the procedure for putting M into normal form by a transformation (9). The other entries will be discussed in a later series of Examples.

For the degenerate parabolic case, begin with the defining equation of M in the form (8),

z_{2} = h_{2}(z_{1}, z̄_{1}) = z_{1}z̄_{1} + \frac{1}{2}(z_{1}^{2} + z̄_{1}^{2}) + e,

where Q = \frac{1}{2}(z_{1} + z̄_{1})^{2}, C ≡ 0 and

(20) e(z_{1}, z̄_{1}) = e_{2}^{04}z̄_{1}^{4} + e_{2}^{13}z̄_{1}^{3}z_{1} + e_{2}^{22}z̄_{1}^{2}z_{1}^{2} + e_{2}^{31}z̄_{1}z_{1}^{3} + e_{2}^{40}z_{1}^{4} + O(5).

The transformations (9) which preserve the quadratic and cubic part of h_{2}, while contributing to the normalization of the quartic terms in e, are of the form

(21)  z̃_{1} = a_{1}z_{1} + a_{2}z_{2} + p_{1}^{1}z_{1}^{3} + p_{1}^{2}z_{1}z_{2}
z̃_{2} = (a_{1})^{2}z_{2} + p_{2}^{1}z_{1}^{4} + p_{2}^{2}z_{1}^{2}z_{2} + p_{2}^{3}z_{2}^{2},

where a_{1} is a nonzero real number and a_{2} is purely imaginary. Higher weight terms in the transformation would only contribute degree 5 or higher quantities to the transformed defining equation.

z̃_{2} − \frac{1}{2}(z̃_{1} + ¯ z̃_{1})^{2}

= (a_{1})^{2}z_{2} + p_{2}^{1}z_{1}^{4} + p_{2}^{2}z_{1}^{2}z_{2} + p_{2}^{3}z_{2}^{2}

−\frac{1}{2}(a_{1}z_{1} + a_{2}z_{2} + p_{1}^{1}z_{1}^{3} + p_{1}^{2}z_{1}z_{2} + \overline{(a_{1}z_{1} + a_{2}z_{2} + p_{1}^{1}z_{1}^{3} + p_{1}^{2}z_{1}z_{2})})^{2},

and for points x = (z_{1}, z_{2}) on M near 0, substituting z_{2} = h_{2} gives:

z̃_{1}
(22) = a_{1}z_{1} + a_{2}(\frac{1}{2}(z_{1} + z̄_{1})^{2} + e) + p_{1}^{1}z_{1}^{3} + p_{1}^{2}z_{1} · (\frac{1}{2}(z_{1} + z̄_{1})^{2} + e),

z̃_{2} − \frac{1}{2}(z̃_{1} + ¯ z̃_{1})^{2} (23)

= (a_{1})^{2}(e_{2}^{04}z̄_{1}^{4} + e_{2}^{13}z̄_{1}^{3}z_{1} + e_{2}^{22}z̄_{1}^{2}z_{1}^{2} + e_{2}^{31}z̄_{1}z_{1}^{3} + e_{2}^{40}z_{1}^{4})

+p_{2}^{1}z_{1}^{4} + p_{2}^{2}z_{1}^{2} \frac{1}{2}(z_{1} + z̄_{1})^{2} + p_{2}^{3}(\frac{1}{2}(z_{1} + z̄_{1})^{2})^{2}

−a_{1}(z_{1} + z̄_{1}) · (p_{1}^{1}z_{1}^{3} + p_{1}^{2}z_{1} \frac{(z_{1} + z̄_{1})^{2}}{2} + \overline{p_{1}^{1}}z̄_{1}^{3} + \overline{p_{1}^{2}}z̄_{1} \frac{(z_{1} + z̄_{1})^{2}}{2} )

+O(5).

Substituting the inverse function of (22), z_{1} = \frac{1}{a_{1}} z̃_{1} + O(2), into (23) gives the new defining equation in the z̃ coordinates:

(24) z̃_{2} = \frac{1}{2}(z̃_{1} + ¯ z̃_{1})^{2}

+\frac{1}{(a_{1})^{2}} (e_{2}^{04} ¯z̃_{1}^{4} + e_{2}^{13} ¯z̃_{1}^{3}z̃_{1} + e_{2}^{22} ¯z̃_{1}^{2}z̃_{1}^{2} + e_{2}^{31} ¯z̃_{1}z̃_{1}^{3} + e_{2}^{40}z̃_{1}^{4})

+\frac{1}{(a_{1})^{4}} (p_{2}^{1}z̃_{1}^{4} + p_{2}^{2}z̃_{1}^{2} \frac{1}{2}(z̃_{1} + ¯ z̃_{1})^{2} + p_{2}^{3}(\frac{1}{2}(z̃_{1} + ¯ z̃_{1})^{2})^{2})

−\frac{z̃_{1} + ¯ z̃_{1}}{(a_{1})^{3}} · (p_{1}^{1}z̃_{1}^{3} + p_{1}^{2}z̃_{1} \frac{(z̃_{1} + ¯ z̃_{1})^{2}}{2} + \overline{p_{1}^{1}} ¯z̃_{1}^{3} + \overline{p_{1}^{2}} ¯z̃_{1} \frac{(z̃_{1} + ¯ z̃_{1})^{2}}{2} )

+O(5).

Observe that the imaginary coefficient a_{2} does not contribute to the degree 4 terms. The normal form problem is, given e, find p⃗ which simplifies (24). This is a (real) linear problem and a computer algebra system ([Maple]) is useful; the remainder of the calculation is broken into steps and just sketched here. The first step is that from the form of (24), it appears that one could find p_{2}^{1}, p_{2}^{2}, p_{2}^{3} that eliminate the complex coefficients e_{2}^{04}, e_{2}^{22}, e_{2}^{40}, while p_{1}^{1}, p_{1}^{2} are chosen to be 0 and a_{1} = 1. In fact, the following choice works:

p_{2}^{3} = −4e_{2}^{04}, p_{2}^{2} = −2e_{2}^{22} + 12e_{2}^{04}, p_{2}^{1} = e_{2}^{22} − e_{2}^{40} − 5e_{2}^{04}.

Applying such a transformation, and then dropping the tilde notation, gets M in the form (20), but with e_{2}^{04} = e_{2}^{22} = e_{2}^{40} = 0, and the new complex coefficients e_{2}^{13}, e_{2}^{31} possibly changed from the old ones.

The next transformation will be another with p_{1}^{2} = 0, a_{1} = 1, but will use p_{1}^{1} to normalize e_{2}^{13} and e_{2}^{31}. To preserve the partial normal form e_{2}^{04} = e_{2}^{22} = e_{2}^{40} = 0, the coefficients of p_{2} must be

p_{2}^{2} = −12\overline{p_{1}^{1}}, p_{2}^{1} = 5\overline{p_{1}^{1}} + p_{1}^{1}, p_{2}^{3} = 4\overline{p_{1}^{1}}.

Then (24) becomes:

z_{2} = Q + (3\overline{p_{1}^{1}} + e_{2}^{13})z_{1}z̄_{1}^{3} + (−p_{1}^{1} + e_{2}^{31} − 8\overline{p_{1}^{1}})z_{1}^{3}z̄_{1} + O(5),

and for any e_{2}^{13}, e_{2}^{31}, there is some p_{1}^{1} so that the above coefficients become equal.

The third transformation will use a_{1} and all the unknown p⃗ coefficients, and again, to preserve the partial normal form e_{2}^{04} = e_{2}^{22} = e_{2}^{40} = 0, e_{2}^{13} = e_{2}^{31}, the coefficients of p_{1} and p_{2} must satisfy:

p_{1}^{1} = −\frac{1}{5}(p_{1}^{2} − \overline{p_{1}^{2}}),
p_{2}^{1} = −\frac{1}{5}a_{1}(p_{1}^{2} − \overline{p_{1}^{2}}),
p_{2}^{2} = \frac{3}{5}a_{1}(p_{1}^{2} − \overline{p_{1}^{2}}),
p_{2}^{3} = \frac{2}{5}a_{1}(2p_{1}^{2} + 3\overline{p_{1}^{2}}).

This brings (24) to:

z_{2} = Q + (\frac{e_{2}^{13}}{(a_{1})^{2}} + \frac{p_{1}^{2}}{10(a_{1})^{3}} − \frac{\overline{p_{1}^{2}}}{10(a_{1})^{3}} ) (z_{1}^{3}z̄_{1} + z_{1}z̄_{1}^{3}) + O(5).

Only the imaginary part of p_{1}^{2} contributes to these terms, so e_{2}^{13} can be normalized to be a real number, and then the positive scale factor (a_{1})^{2} will make e_{2}^{13} either +1, −1, or 0, as claimed.

The classification of CR singularities of surfaces in ℂ^{2} into elliptic, parabolic, and hyperbolic cases is well known, but a few brief remarks follow; for recent survey articles, see [BER], [S-G].

Example 5.2. The higher degree terms for the generic elliptic case were considered by [MW], who showed that M has a real algebraic implicit normal form (to be recalled in Example 5.13).

Example 5.3. The γ = 0, s = 3, 4, … cases were considered by [MW] and [Moser], and also by [HK], who proved that there exists a holomorphic transformation so that h_{2} is real valued, and M fits inside ℝ^{3} ⊆ ℂ^{2}. A real surface contained in ℝ^{3} is said to be “holomorphically flat,” and the transformation of a surface into a flat normal form is called “flattening.” The term “cusp” describing the CR singularity is borrowed here from the terminology of singularity theory, since the s = 3 form of h_{2} bears a resemblance to the normal form for Whitney’s cusp singularity of a map ℂ^{2} → ℂ^{2} : (z_{1}, ζ_{1}) ↦ (z_{1}, z_{1}ζ_{1} + ζ_{1}^{3}) ([AVGL] §3.1, [GG] §VI.2, [Lu] §2.6, [Martinet], [Wall]). The resemblance becomes more concrete in view of the complexification construction of Section 4.

Example 5.4. The γ = 0, s = ∞ case was considered by [Moser], who showed that if h_{2} can be transformed to z_{1}z̄_{1} + O(s) for all s = 3, 4, …, then there exists a holomorphic coordinate change transforming M into the real quadric variety.

Example 5.5. Cubic normal forms for both the nondegenerate and degenerate parabolic cases were found by [Webster_{2}]. In the nondegenerate case, the normal form for C given by [Webster_{2}] is C = −iz_{1}z̄_{1}(z_{1} + z̄_{1}), however it is possible to choose the signs differently so that the Proposition’s normal form gives a real valued Q + C.

Example 5.6. In most of the hyperbolic cases, with γ > \frac{1}{2}, the cubic terms can be eliminated just as in the generic elliptic case. However, it was shown by [MW] that for γ satisfying a certain diophantine condition, M cannot always be flattened into ℝ^{3} by making the higher degree terms real valued. The only value of γ for which this happens at the cubic terms is γ = 1, where the coefficient of the imaginary quantity (z_{1} − z̄_{1})z_{1}z̄_{1} can be normalized to either 1 or 0.

Example 5.7. The normal form Q = z_{1}^{2} + z̄_{1}^{2} is considered the γ → +∞ limit of the hyperbolic normal forms. An alternative normal form would be the complex valued monomial Q = z̄_{1}^{2}. The label “fold” again comes from its similarity to Whitney’s normal form for the fold singularity, ℂ^{2} → ℂ^{2} : (z_{1}, ζ_{1}) ↦ (z_{1}, ζ_{1}^{2}).

The presence of the continuous invariant γ interpolating between the fold and cusp normal forms is the most obvious way in which the classification problem for CR singularities of surfaces in ℂ^{2} differs from the analogous problem for singularities of maps ℂ^{2} → ℂ^{2}.

For purposes of comparison with the n = 2 case, and also with the normal forms under u-equivalence in the next Subsection, the following table of normal forms for n-submanifolds of ℂ^{n}, n ≥ 3, is recalled from [MW] and [Webster_{2}].

Proposition 5.8. For n ≥ 3 and any real analytic n-submanifold M ⊆ ℂ^{n} with a CR singularity x ∈ N_{1} \ N_{2}, there is a normal form {z_{n} = h_{n}(z_{1}, z̄_{1}, x), y_{σ} = H_{σ}(z_{1}, z̄_{1}, x), σ = 2, …, n − 1} from exactly one row in the following list equivalent to M under a holomorphic transformation of a neighborhood of x.

Label  normal form  comment

elliptic generic

\begin{matrix}h_{n} = z_{1}z̄_{1} + γ(z_{1}^{2} + z̄_{1}^{2}) + O(3) real \\ H_{σ} ≡ 0 \end{matrix} 0 < γ < \frac{1}{2}

elliptic cusp

\begin{matrix}h_{n} = z_{1}z̄_{1} + O(3) \\ H_{σ} = O(3) \end{matrix} γ = 0

parabolic nondegenerate

\begin{matrix}h_{n} = z_{1}z̄_{1} + \frac{1}{2}(z_{1}^{2} + z̄_{1}^{2}) + ic_{n}(z_{1} − z̄_{1})x_{2} \\ −i(z_{1} + z̄_{1})z_{1}z̄_{1} + η_{n}^{σ}x_{σ}z_{1}z̄_{1} + O(4) \\ H_{σ} = O(4) \end{matrix} \begin{matrix}γ = \frac{1}{2} \\ c_{n} = 0, 1 \\ η_{n}^{σ} real \end{matrix}

parabolic degenerate

\begin{matrix}h_{n} = z_{1}z̄_{1} + \frac{1}{2}(z_{1}^{2} + z̄_{1}^{2}) + ic_{n}(z_{1} − z̄_{1})x_{2} \\ +η_{n}^{σ}x_{σ}z_{1}z̄_{1} + O(4) \\ H_{σ} = O(4) \end{matrix} \begin{matrix}γ = \frac{1}{2} \\ c_{n} = 0, 1 \\ η_{n}^{σ} real \end{matrix}

hyperbolic generic

\begin{matrix}h_{n} = z_{1}z̄_{1} + γ(z_{1}^{2} + z̄_{1}^{2}) + O(3) \\ H_{σ} = O(3) \end{matrix} γ > \frac{1}{2}

hyperbolic fold

\begin{matrix}h_{n} = z_{1}^{2} + z̄_{1}^{2} + O(3) \\ H_{σ} = b_{σ}z_{1}z̄_{1} + O(3) \end{matrix} \begin{matrix}γ = ∞ \\ b_{σ} = 0, 1 \end{matrix}

\begin{matrix}h_{n} = (z_{1} + z̄_{1})x_{2} + i(z_{1} − z̄_{1})x_{3} + O(3) \\ H_{σ} = b_{σ}z_{1}z̄_{1} + O(3) \end{matrix}  \begin{matrix}n ≥ 4 \\ b_{σ} = 0, 1 \end{matrix}
\begin{matrix}h_{n} = (z_{1} + z̄_{1})x_{2} + O(3) \\ H_{σ} = b_{σ}z_{1}z̄_{1} + O(3) \end{matrix}  b_{σ} = 0, 1
\begin{matrix}h_{n} = O(3) \\ H_{σ} = b_{σ}z_{1}z̄_{1} + O(3) \end{matrix}  b_{σ} = 0, 1

The quantities γ and c_{n} are biholomorphic invariants.

The constants b_{σ}, η_{n}^{σ} in the above table are not necessarily invariants and could be normalized by the R_{(n−2)×(n−2)} block of entries in (9).

Remark 5.9. The parabolic points of a submanifold M^{n} in general position in ℂ^{n}, n ≥ 3, are characterized by [Webster_{2}] as the points x in N_{1} \ N_{2} where the real tangent space T_{x}N_{1} intersects the complex tangent line T_{x}M ∩ iT_{x}M in a real line, the “parabolic line.” At the elliptic and hyperbolic points, these subspaces of T M intersect only at the origin. This property of parabolic CR singularities bears some resemblance to the S_{r,s} system of classification of singularities of maps, as in [GG] §VI.4, but no connection will be pursued here.

5.2. Unfolding CR singularities of surfaces.

The m = n = 2 case of (10) is that ; M is a (2 + k)-submanifold of ℂ^{2+k}:

(25)  z_{2} = h_{2}(z_{1}, z̄_{1}, t) = O(2)
s_{1} = … = s_{k} = 0,

and by a holomorphic transformation in z_{1}, z_{2} only (leaving w⃗ fixed), we can assume that the quantity h_{2}(z_{1}, z̄_{1}, 0) is in one of the normal forms from Proposition 5.1. So, the quadratic part of h_{2} is

(26)  Q(z_{1}, z̄_{1}, 0) + e_{2}^{10α}t_{α}z_{1} + e_{2}^{01α}t_{α}z̄_{1} + e_{2}^{αβ}t_{α}t_{β},

with complex coefficients summed over α, β = 1, …, k. The group of transformations U_{2,2,k} (11) is:

(27)  z̃_{1} = a_{1}z_{1} + a_{2}z_{2} + a^{α}w_{α} + p_{1}(z_{1}, z_{2}, w⃗)
z̃_{2} = a_{22}z_{2} + p_{2}(z_{1}, z_{2}, w⃗)
(28)  w̃ = R_{k×k}w⃗ + P⃗ (w⃗).

Our approach to the u-equivalence problem is to find some normal forms for (25) under this group. The only differences between this problem and the normal form problem of Proposition 5.8 are that the functions H_{σ} are already identically zero in (25), and the full group from Proposition 5.8 would replace (28) by:

w̃_{α} = r_{α}^{β}w_{β} + a′′_{α}z_{2} + P_{α}(z_{1}, z_{2}, w⃗),

allowing complex a′′_{α} and complex valued P_{α}.

The calculation proving the following result appears in [Bishop].

Lemma 5.10. Given ; M ⊆ ℂ^{2+k} of the form (25), with

h_{2} = z_{1}z̄_{1}+γ(z_{1}^{2}+z̄_{1}^{2})+e_{2}^{10α}t_{α}z_{1}+e_{2}^{01α}t_{α}z̄_{1}+e_{2}^{αβ}t_{α}t_{β} +C(z_{1}, z̄_{1}, t)+O(4),

with γ ≥ 0 and γ ≠ \frac{1}{2}, there exists a transformation of the form

z̃_{1} = z_{1} + a^{α}w_{α}, z̃_{2} = z_{2} + p_{2}^{α}z_{1}w_{α} + p_{2}^{αβ}w_{α}w_{β}, w̃ = w⃗,

such that the new defining equation is

h_{2}(z̃_{1}, ¯ z̃_{1}, t̃) = z̃_{1} ¯z̃_{1} + γ(z̃_{1}^{2} + ¯ z̃_{1}^{2}) + C(z̃_{1}, ¯ z̃_{1}, t̃) + O(4),

where the corresponding coefficients of C(z̃_{1}, ¯ z̃_{1}, 0) are the same as those of C(z_{1}, z̄_{1}, 0).

This means that in the elliptic or hyperbolic cases, the quadratic terms involving t can always be eliminated by an element of the group (27–28), without altering the cubic terms in z_{1}, z̄_{1} only. An analogous result holds for the γ = ∞ case.

The γ = \frac{1}{2} case is considered in Example 5.18, but for any quadratic part Q(z_{1}, z̄_{1}, 0) in (26), a transformation in the group (27–28) of the form z̃_{2} = z_{2} + p_{2}^{α}z_{1}w_{α} + p_{2}^{αβ}w_{α}w_{β} will eliminate, or re-assign any complex value to, the coefficients e_{2}^{10α}, e_{2}^{αβ} in (26), without changing any other quadratic terms in h_{2}. More generally, terms in h_{2} of the form e_{2}^{a0K}z_{1}^{a}t^{K}, without any z̄_{1} factor, can be assigned any complex coefficient by a transformation of the form z̃_{2} = z_{2} + p_{2}^{aK}z_{1}^{a}w^{K} without changing any other terms of the same or lower degree. So, the defining equation (25) of ; M is of the form h_{2} = Q(z_{1}, z̄_{1}, t) + C(z_{1}, z̄_{1}, t) + e(z_{1}, z̄_{1}, t), where

C = C(z_{1}, z̄_{1}, 0) + e_{2}^{20α}z_{1}^{2}t_{α} + e_{2}^{11α}z_{1}z̄_{1}t_{α} + e_{2}^{02α}z̄_{1}^{2}t_{α}
+e_{2}^{10αβ}z_{1}t_{α}t_{β} + e_{2}^{01αβ}z̄_{1}t_{α}t_{β} + e_{2}^{00K}t^{K},

and the coefficients e_{2}^{20α}, e_{2}^{10αβ}, e_{2}^{00K} (where K is a degree three multiindex) can take any value after a weight 3 holomorphic transformation of z_{2}. However, the normalization of the other cubic coefficients depends on the quadratic part Q(z_{1}, z̄_{1}, t) and on C(z_{1}, z̄_{1}, 0). The rest of this Subsection will consider a series of Examples of the most generic u-equivalence classes of unfoldings of the CR singularities with the various normal forms Q+C from Proposition 5.1. For the sake of economy, we will assume the e_{2}^{αβ}t_{α}t_{β} and e_{2}^{00K}t^{K} terms are eliminated at the beginning of each calculation and also at each step where they might re-appear, although we keep the terms e_{2}^{10α}z_{1}t_{α}, e_{2}^{20α}z_{1}^{2}t_{α}, e_{2}^{10αβ}z_{1}t_{α}t_{β}, since they may be used later to get a real valued normal form.

Example 5.11. For M = M_{0} with a generic elliptic or generic hyperbolic singularity, with C(z_{1}, z̄_{1}, 0) = 0 as in Proposition 5.1, Lemma 5.10 applies to ; M, and (25) becomes h_{2} = Q+C(z_{1}, z̄_{1}, t)+O(4), where

(29) C = e_{2}^{20α}z_{1}^{2}t_{α} + e_{2}^{11α}z_{1}z̄_{1}t_{α} + e_{2}^{02α}z̄_{1}^{2}t_{α} + e_{2}^{10αβ}z_{1}t_{α}t_{β} + e_{2}^{01αβ}z̄_{1}t_{α}t_{β}.

The transformations that could alter the cubic terms in C while preserving the normal form of h_{2}(z_{1}, z̄_{1}, 0) are:

(30)  z̃_{1} = a_{1}z_{1} + p_{1}^{1α}z_{1}w_{α} + p_{1}^{αβ}w_{α}w_{β}
z̃_{2} = (a_{1})^{2}z_{1} + p_{2}^{1α}z_{1}^{2}w_{α} + p_{2}^{2α}z_{2}w_{α} + p_{2}^{1αβ}z_{1}w_{α}w_{β}
w̃ = Rw⃗,

where a_{1} is a nonzero real scalar, and higher weight terms in p⃗, that would not contribute cubic terms in h_{2}, are omitted. Without stating all the details here, the coefficients p_{1}^{αβ}, p_{2}^{1αβ} can eliminate the terms e_{2}^{01αβ}z̄_{1}t_{α}t_{β}, e_{2}^{10αβ}z_{1}t_{α}t_{β} (the cancellation is similar to that of Lemma 5.10). We choose to use coefficient p_{2}^{2α} to eliminate e_{2}^{11α}z_{1}z̄_{1}t_{α}, and to use p_{2}^{1α}z_{1}^{2}w_{α} to equate the coefficients of e_{2}^{20α}z_{1}^{2}t_{α} and e_{2}^{02α}z̄_{1}^{2}t_{α}. The stabilizer of the partial normal form

C = e_{2}^{02α}(z_{1}^{2} + z̄_{1}^{2})t_{α}

is of the form (z̃, w̃) as in (30), with p_{1}^{αβ} = p_{2}^{1αβ} = 0, p_{2}^{2α} = a_{1}(p_{1}^{1α}+\overline{p_{1}^{1α}}), p_{2}^{1α} = 2a_{1}β(p_{1}^{1α} − \overline{p_{1}^{1α}}). With R equal to the identity matrix 𝟙, the transformed quantity C is

(e_{2}^{02α} + \frac{γ}{a_{1}} (p_{1}^{1α} − \overline{p_{1}^{1α}})) (z_{1}^{2} + z̄_{1}^{2})t_{α}.

So, each p_{1}^{1α} only contributes its imaginary part, and e_{2}^{02α} can be normalized to be real. Unless all the e_{2}^{02α} are zero, the real matrix R then can transform the vector of coefficients to (1, 0, …, 0), so there is a generic cubic normal form:

(31)  z_{2} = z_{1}z̄_{1} + (γ + t_{1}) · (z_{1}^{2} + z̄_{1}^{2}) + O(4),

and also a degenerate case where the quadratic and cubic parts do not depend on t.

Geometrically, the unfolding is about what one would expect: manifolds M_{t} near M_{0} will have Bishop invariant varying near γ, and (at least up to some degree) this variation can be normalized to depend linearly only on one t coordinate. For t close to 0, if M_{0} is elliptic, then so is M_{t}, and similarly if M_{0} is hyperbolic, then so is M_{t}. Since only the linear part of the transformation of the w⃗ coordinates contributed to the normalization, there is no difference in appearance between this cubic normal form under u-equivalence, and the normal form from Proposition 5.8, for n-manifolds in ℂ^{n} under the larger transformation group (where n = 2 + k).

Example 5.12. The diophantine cases from Proposition 5.1 have unfoldings similar to those in the previous Example. In the degenerate case, where γ = 1, C = 0, the calculations of the previous Example work without changing the result. In the nondegenerate case, the subgroup (30) preserving the cubic part h_{2}(z_{1}, z̄_{1}, 0) = Q + (z_{1} − z̄_{1})z_{1}z̄_{1} + O(4) is different, in particular, a_{1} must equal 1, but the end result is the same, and the generic unfolding has normal form

z_{2} = z_{1}z̄_{1} + (1 + t_{1}) · (z_{1}^{2} + z̄_{1}^{2}) + (z_{1} − z̄_{1})z_{1}z̄_{1} + O(4).

Example 5.13. In the generic elliptic case, the holomorphic invariants in the higher degree terms were found by [MW]; the implicit normal form is the polynomial defining equation

z_{2} = z_{1}z̄_{1} + (γ + δx_{2}^{s}) · (z_{1}^{2} + z̄_{1}^{2}),

where 0 < γ < \frac{1}{2}, δ = ±1, and the integer s = 1, 2, 3, … (or s = ∞ in the quadric case) determine the local biholomorphic equivalence class of a real analytic surface. When put into the graph form (8), the defining equation for M is

(32) z_{2} = z_{1}z̄_{1} + (γ + δ̂(z_{1}z̄_{1} + γ(z_{1}^{2} + z̄_{1}^{2}))) · (z_{1}^{2} + z̄_{1}^{2}) + O(6),

where δ̂ = δ if s = 1, and δ̂ = 0 if s > 1 or s = ∞. A nondegenerate unfolding of this surface will be of the form (25), with

h_{2} = Q + C + δ̂(z_{1}z̄_{1} + γ(z_{1}^{2} + z̄_{1}^{2})) · (z_{1}^{2} + z̄_{1}^{2}) + F (z_{1}, z̄_{1}, t) + O(5),

where Q + C is as in (31), and the degree 4 terms depending on t are

(33)  F = e_{2}^{30α}z_{1}^{3}t_{α} + e_{2}^{21α}z_{1}^{2}z̄_{1}t_{α} + e_{2}^{12α}z_{1}z̄_{1}^{2}t_{α} + e_{2}^{03α}z̄_{1}^{3}t_{α}
+e_{2}^{20αβ}z_{1}^{2}t_{α}t_{β} + e_{2}^{11αβ}z_{1}z̄_{1}t_{α}t_{β} + e_{2}^{02αβ}z̄_{1}^{2}t_{α}t_{β}
+e_{2}^{10K}z_{1}t^{K} + e_{2}^{01K}z̄_{1}t^{K}.

A transformation (27–28) of the form

(34) z̃_{1} = z_{1} + p_{1}^{20α}z_{1}^{2}w_{α} + p_{1}^{01α}z_{2}w_{α} + p_{1}^{1αβ}z_{1}w_{α}w_{β} + p_{1}^{K}w^{K}
(35) z̃_{2} = z_{2} + p_{2}^{30α}z_{1}^{3}w_{α} + p_{2}^{11α}z_{1}z_{2}w_{α}
+p_{2}^{20αβ}z_{1}^{2}w_{α}w_{β} + p_{2}^{01αβ}z_{2}w_{α}w_{β} + p_{2}^{K}z_{1}w^{K}
w̃_{1} = w_{1} + P_{1}^{αβ}w_{α}w_{β}
w̃_{j} = w_{j}, j = 2, …, k,

where P_{1}^{αβ} are real coefficients and p_{1}^{1αβ} are purely imaginary, can transform F to zero. Again, without stating all the details, the coefficients p_{1}^{K}, p_{2}^{K} eliminate e_{2}^{01K} and e_{2}^{10K}, the coefficients p_{2}^{20αβ}, p_{2}^{01αβ} eliminate e_{2}^{20αβ} and e_{2}^{11αβ}, and P_{1}^{αβ}, p_{1}^{1αβ} cancel the real and imaginary parts of e_{2}^{02αβ}. Then p_{1}^{20α}, p_{1}^{01α}, p_{2}^{30α}, p_{2}^{11α} can eliminate e_{2}^{30α}, e_{2}^{21α}, e_{2}^{12α}, e_{2}^{03α}.

The conclusion is that the generic k-parameter unfolding ; M of a surface M with a generic elliptic point (32) has a normal form under the action of the unfolding subgroup:

(36) h_{2} = z_{1}z̄_{1} + (γ + t_{1} + δ̂(z_{1}z̄_{1} + γ(z_{1}^{2} + z̄_{1}^{2}))) · (z_{1}^{2} + z̄_{1}^{2}) + O(5),

which is the same, up to O(5), as the implicit algebraic normal form of [MW] (Equation 5.4, adapted to our notation):

(37)  z_{2} = z_{1}z̄_{1} + (γ + t_{1} + δx_{2}^{s}) · (z_{1}^{2} + z̄_{1}^{2}),

for a real (2 + k)-submanifold in ℂ^{2+k} under the action of local biholomorphisms.

Example 5.14. The next case to be considered is the γ = 0, s = ∞ elliptic normal form, where M = M_{0} has quadratic part Q = z_{1}z̄_{1} and, like the generic elliptic case, zero cubic part. Applying Lemma 5.11, the defining equation (25) becomes h_{2} = z_{1}z̄_{1} +C(z_{1}, z̄_{1}, t)+O(4), where C is the same as (29). The transformations preserving this partial normal form are:

(38)  z̃_{1} = a_{1}z_{1} + a_{2}z_{2} − \frac{a_{1}\overline{a_{2}}}{\overline{a_{1}}} z_{1}^{2} + p_{1}^{α}z_{1}w_{α} + p_{1}^{αβ}w_{α}w_{β}
z̃_{2} = |a_{1}|^{2}z_{1} + p_{2}^{1α}z_{1}^{2}w_{α} + p_{2}^{2α}z_{2}w_{α} + p_{2}^{1αβ}z_{1}w_{α}w_{β}
w̃ = Rw⃗,

where a_{1} is a nonzero complex scalar. As in Example 5.11, the coefficients p_{1}^{αβ}, p_{2}^{1αβ}, p_{2}^{2α} can eliminate the coefficients e_{2}^{01αβ}, e_{2}^{10αβ}, e_{2}^{11α}.

The stabilizer of the new partial normal form is the above group with the conditions p_{2}^{1αβ} = p_{1}^{αβ} = 0, p_{2}^{2α} = a_{1}\overline{p_{1}^{α}} + \overline{a_{1}}p_{1}^{α}, and the new defining equation after such a transformation with R = 𝟙 satisfies:

(39)  C(z_{1}, z̄_{1}, t) = (\frac{p_{2}^{1α}}{(a_{1})^{2}} + \frac{\overline{a_{1}}}{a_{1}} e_{2}^{20α}) z_{1}^{2}t_{α} + \frac{a_{1}}{\overline{a_{1}}} e_{2}^{02α}z̄_{1}^{2}t_{α}.

Note that none of the terms in (38) contributes any quantities of the form z̄_{1}^{2}t_{α}, although each z_{1}^{2}t_{α} can be assigned any coefficient by choice of p_{2}^{1α}.

In the one-parameter (k = 1) case, the single coefficient e_{2}^{021} (= e_{2}^{02α} with α = 1) can be rotated by a_{1} and scaled by R_{1×1}, so for e_{2}^{021} ≠ 0, a normal form for the 3-manifold ; M is

(40)  z_{2} = z_{1}z̄_{1} + (z_{1}^{2} + z̄_{1}^{2})t_{1} + O(4)
s_{1} = 0.

This is simply the γ = 0 case of (31), and geometrically, the circular paraboloid shape with γ = 0 deforms into an elliptical paraboloid with γ = t_{1}.

However, for k ≥ 2, there are k complex coefficients e_{2}^{02α}, but there is still only one complex scalar a_{1}. The real matrix R acts on the real and imaginary parts of the coefficients; it will generically take the vector (e_{2}^{021}, e_{2}^{022}, …, e_{2}^{02k}) to (1, i, 0, …, 0), so a generic real valued normal form is

(41)  z_{2} = z_{1}z̄_{1} + (z̄_{1}^{2} + z_{1}^{2})t_{1} + i(z̄_{1}^{2} − z_{1}^{2})t_{2} + O(4)
s_{α} = 0.

The normal form (40) can be obtained as a non-generic case, for example, if the e_{2}^{02α} are all real.

Example 5.15. Here we consider the elliptic cusp singularity, where the defining equation of M is z_{2} = z_{1}z̄_{1} + z_{1}^{3} + z̄_{1}^{3} + O(4). The calculations from the previous Example go through without much change. In particular, the subgroup (38) is the same, except that to preserve the form of the cubic terms z_{1}^{3} + z̄_{1}^{3}, the complex scalar a_{1} must be a cube root of unity. For k ≥ 2, the matrix R is enough to normalize the generic coefficients e_{2}^{02α} even with just a_{1} = 1, so the generic normal form is similar to (41):

(42) z_{2} = z_{1}z̄_{1} + z_{1}^{3} + z̄_{1}^{3} + (z̄_{1}^{2} + z_{1}^{2})t_{1} + i(z̄_{1}^{2} − z_{1}^{2})t_{2} + O(4)
s_{α} = 0.

However, for k = 1, a nonzero complex coefficient e_{2}^{021} can be scaled to the unit circle by R_{1×1}, but can only be rotated by \sqrt[3]{1} (the factor \frac{a_{1}}{\overline{a_{1}}} from (39)). So, the cubic normal form for a generic one-parameter deformation has a “modulus,” a continuous invariant under the group U_{2,2,1}:

(43) z_{2} = z_{1}z̄_{1} + z_{1}^{3} + z̄_{1}^{3} + (e^{iθ}z̄_{1}^{2} + e^{−iθ}z_{1}^{2})t_{1} + O(4), 0 ≤ θ < \frac{2π}{3}
s_{1} = 0.

Example 5.16. For the higher cusp cases, with γ = 0 and 3 < s < ∞ in the normal forms for M, the nearby manifolds M_{t} are generic elliptic, with a small but nonzero Bishop invariant, except in degenerate cases. In the k = 1 case, there will be a similar root of unity phenomenon in the stabilizer of the degree s normal form.

The generic normal form for the s = 4, k > 1 case is again similar to (41):

(44) z_{2} = z_{1}z̄_{1} + z_{1}^{4} + z̄_{1}^{4} + (z̄_{1}^{2} + z_{1}^{2})t_{1} + i(z̄_{1}^{2} − z_{1}^{2})t_{2} + O(4)
s_{α} = 0

where the O(4) quantity may contain degree 4 terms depending on t. For k = 1, the quartic normal form for a generic one-parameter deformation has a modulus:

(45) z_{2} = z_{1}z̄_{1} + z_{1}^{4} + z̄_{1}^{4} + (e^{iθ}z̄_{1}^{2} + e^{−iθ}z_{1}^{2})t_{1} + O(5), 0 ≤ θ < π
s_{1} = 0.

Without going through the details, the degree 4 terms depending on t, as in (33), can be eliminated in this k = 1 case.

Example 5.17. The hyperbolic γ = ∞ fold singularity has quadratic part Q = z_{1}^{2} + z̄_{1}^{2} and, like the generic elliptic case, zero cubic part. As mentioned earlier, a version of Lemma 5.10 works, so that the terms e_{2}^{10α}z_{1}t_{α} + e_{2}^{01α}z̄_{1}t_{α} can be eliminated without introducing any cubic terms in z_{1}, z̄_{1} only. The defining equation (25) becomes h_{2} = z_{1}z̄_{1} + C(z_{1}, z̄_{1}, t) + O(4), where C is the same as (29). The transformations preserving this partial normal form are:

(46) z̃_{1} = a_{1}z_{1} + p_{1}^{α}z_{1}w_{α} + p_{1}^{αβ}w_{α}w_{β}
z̃_{2} = \overline{a_{1}}^{2}z_{1} + ((a_{1})^{2} − \overline{a_{1}}^{2})z_{1}^{2} + p_{2}^{1α}z_{1}^{2}w_{α} + p_{2}^{2α}z_{2}w_{α} + p_{2}^{1αβ}z_{1}w_{α}w_{β}
w̃ = Rw⃗,

where a_{1} is a nonzero complex scalar. The coefficients p_{1}^{αβ}, p_{2}^{1αβ}, p_{2}^{1α}, p_{2}^{2α} can eliminate the coefficients e_{2}^{01αβ}, e_{2}^{10αβ}, e_{2}^{20α}, e_{2}^{02α}.

The stabilizer of the new partial normal form is the above group with the conditions p_{2}^{2α} = 2\overline{a_{1}p_{1}^{α}}, p_{2}^{1α} = 2a_{1}p_{1}^{α} − 2\overline{a_{1}p_{1}^{α}}, and the new defining equation after such a transformation with R = 𝟙 satisfies:

C(z_{1}, z̄_{1}, t) = \frac{\overline{a_{1}}}{a_{1}} e_{2}^{11α}z_{1}z̄_{1}t_{α}.

Note that none of the terms in (46) contributes any quantities of the form z_{1}z̄_{1}t_{α}.

In the one-parameter (k = 1) case, the single coefficient e_{2}^{021} (= e_{2}^{02α} with α = 1) can be rotated by a_{1} and scaled by R_{1×1}, so for e_{2}^{021} ≠ 0, a normal form for the 3-manifold ; M is

(47)  z_{2} = z_{1}^{2} + z̄_{1}^{2} + t_{1}z_{1}z̄_{1} + O(4)
s_{1} = 0.

For k ≥ 2, there are k complex coefficients e_{2}^{02α}, but there is still only one complex scalar a_{1}. As in Example 5.14, the real matrix R acts on the real and imaginary parts of the coefficients, generically taking the vector (e_{2}^{021}, e_{2}^{022}, …, e_{2}^{02k}) to (1, i, 0, …, 0), so a generic normal form is

(48)  z_{2} = z̄_{1}^{2} + z_{1}^{2} + (t_{1} + it_{2})z_{1}z̄_{1} + O(4)
s_{α} = 0.

The normal form (47) can be obtained as a non-generic case, for example, if the e_{2}^{02α} are all real.

Example 5.18. The nondegenerate parabolic CR singularity is an important case; it is well-known that it is an unstable singularity for a surface M = M_{0} in ℂ^{2}, in the sense that, unlike the elliptic or hyperbolic properties from Example 5.11, one expects that surfaces M_{t} near M will generally not all have parabolic singularities. The following unfolding calculation makes this more precise.

For M ⊆ ℂ^{2} defined by z_{2} = Q + C + O(4) with quadratic normal form Q = z_{1}z̄_{1} + \frac{1}{2}(z_{1}^{2} + z̄_{1}^{2}), the transformations (9) of ℂ^{2} stabilizing the quadratic normal form are z̃_{1} = a_{1}z_{1} +p_{1}, z̃_{2} = (a_{1})^{2}z_{2} +p_{2}, with p_{1} weight 2, p_{2} weight 3, and a_{1} real and nonzero. For C = i(z_{1} − z̄_{1})z_{1}z̄_{1} as in Proposition 5.1, the stabilizer of the cubic normal form is z̃_{1} = z_{1} + a_{2}z_{2} + p_{1}, z̃_{2} = z_{2} + p_{2}, with a_{1} = 1, a_{2} purely imaginary, p_{1} weight 3, and p_{2} weight 4.

For ; M ⊆ ℂ^{2+k} as in (26), a transformation (27–28) of the form z̃_{1} = a_{1}z_{1}+a^{α}w_{α}, z̃_{2} = (a_{1})^{2}z_{2}, w̃ = w⃗ transforms the term e_{2}^{01α}z̄_{1}t_{α} into (a_{1}e_{2}^{01α} − a^{α} − \overline{a^{α}})z̄_{1}t_{α}. So, unlike every one of the previous Examples, only the real part of a^{α} contributes to the normalization. Canceling the real parts of e_{2}^{01α} leaves a vector of purely imaginary coefficients, which is either the zero vector, or can be normalized by R_{k×k} to −i · (1, 0, …, 0). Then, a transformation z̃_{2} = z_{2} + p_{2}^{α}z_{1}w_{α} gives a real valued quadratic part for the following normal form for ; M:

(49) z_{2} = z_{1}z̄_{1} + \frac{1}{2}(z_{1}^{2} + z̄_{1}^{2}) + i(z_{1} − z̄_{1})t_{1} + C(z_{1}, z̄_{1}, t) + O(4),

where C(z_{1}, z̄_{1}, 0) = i(z_{1} − z̄_{1})z_{1}z̄_{1} is as in Proposition 5.1 and the other terms in C are as in (29). This normal form already resembles the normal form for a parabolic CR singularity of a real n-manifold in ℂ^{n}, as in Proposition 5.8.

The transformation group (27–28) contributing to the normalization of cubic terms, without introducing quadratic terms, is

(50)  z̃_{1} = z_{1} + a_{2}z_{2} + a^{α}w_{α} + p_{1}(z_{1}, z_{2}, w⃗)
z̃_{2} = z_{2} + p_{2}^{1α}w_{1}w_{α} + p_{2}(z_{1}, z_{2}, w⃗)
w̃_{1} = w_{1} + P_{1}(w⃗)
w̃_{j} = r_{j}^{α}w_{α} + P_{j}(w⃗), j = 2, …, k

with a_{2} and a^{α} purely imaginary and p_{2}^{1α} = 2ia^{α}.

If we use a particular transformation of the above form with a_{2} = 0, p_{1} = p_{1}^{α}z_{1}w_{α}, p_{2} = p_{2}^{α}z_{2}w_{α}, w̃ = w⃗, the coefficient of e_{2}^{02α}z̄_{1}^{2}t_{α} is transformed to

e_{2}^{02α} + \frac{1}{2}p_{2}^{α} − \overline{p_{1}^{α}} + ia^{α},

and the coefficient of e_{2}^{11α}z_{1}z̄_{1}t_{α} is transformed to

e_{2}^{11α} + p_{2}^{α} − p_{1}^{α} − \overline{p_{1}^{α}} − 4ia^{α}.

The a^{α} quantities come from transforming the cubic term i(z_{1}−z̄_{1})z_{1}z̄_{1}, so this step is where the nondegeneracy property of M is used. Setting both quantities to zero gives a system of two equations that has a solution: solving the first for p_{2}^{α} and plugging into the second gives

\frac{1}{2}(p_{1}^{α} − \overline{p_{1}^{α}}) + 3ia^{α} − \frac{1}{2}e_{2}^{11α} + e_{2}^{02α} = 0,

where the imaginary part of p_{1}^{α} and the real quantity 3ia^{α} are enough to cancel the given complex part. This cancellation, using the linear coefficients a^{α}, takes the normalization a little further than that of [Webster_{2}]; in the nondegenerate parabolic case of Proposition 5.8, the coefficients η_{n}^{σ} can be canceled by a transformation of ℂ^{n} in B_{n,n}.

The previous step cancels the e_{2}^{02α}z̄_{1}^{2}t_{α} and e_{2}^{11α}z_{1}z̄_{1}t_{α} terms, but also changes the coefficients of z̄_{1}t_{α}t_{β} and other terms. We reassign the same e_{2} labels to the new coefficients. Another particular transformation,

z̃_{1} = z_{1} + p_{1}^{αβ}w_{α}w_{β}
z̃_{2} = z_{2} + p_{2}^{1αβ}z_{1}w_{α}w_{β}
w̃_{1} = w_{1} + P_{1}^{αβ}w_{α}w_{β}
w̃_{j} = w_{j}, j = 2, …, k

transforms the coefficient of e_{2}^{01αβ}z̄_{1}t_{α}t_{β} to

e_{2}^{01αβ} − (p_{1}^{αβ} + \overline{p_{1}^{αβ}}) + iP_{1}^{αβ},

without reintroducing either of the previously normalized terms z_{1}z̄_{1}t_{α}, z̄_{1}^{2}t_{α}. The real part of p_{1}^{αβ} and the real coefficients P_{1}^{αβ} can eliminate all the e_{2}^{01αβ}z̄_{1}t_{α}t_{β} terms. The coefficients p_{2}^{1αβ} can eliminate the e_{2}^{10αβ}z_{1}t_{α}t_{β} terms and similarly, the remaining terms, not involving z̄_{1}, can be eliminated by transformations of z_{2}, so the generic normal form for ; M has cubic part not depending on t:

z_{2} = z_{1}z̄_{1} + \frac{1}{2}(z_{1}^{2} + z̄_{1}^{2}) + i(z_{1} − z̄_{1})t_{1} + i(z_{1} − z̄_{1})z_{1}z̄_{1} + O(4).

The last step with P_{1}^{αβ} required the nondegeneracy of the quadratic normal form (49).

Although the case where the unfolding ; M has a degenerate normal form, in which the quadratic terms do not depend on t, is not considered here, it is notable that the one-parameter unfolding resembling (31):

z_{2} = z_{1}z̄_{1} + (\frac{1}{2} + t_{1}) · (z_{1}^{2} + z̄_{1}^{2}) + i(z_{1} − z̄_{1})z_{1}z̄_{1} + O(4),

is not u-equivalent to a nondegenerate unfolding of the form (49), since by the above calculation, the cubic terms (z_{1}^{2} + z̄_{1}^{2})t_{1} can be eliminated without introducing any quadratic terms.

Example 5.19. For M ⊆ ℂ^{2} with a degenerate parabolic CR singularity, defined by

z_{2} = z_{1}z̄_{1} + \frac{1}{2}(z_{1}^{2} + z̄_{1}^{2}) + η(z_{1}^{2} + z̄_{1}^{2})z_{1}z̄_{1} + O(5),

the normal form has zero cubic part, and the quartic terms are η(z_{1}^{2} + z̄_{1}^{2})z_{1}z̄_{1}, η = ±1, 0. The calculations for a normal form for ; M proceed at first in the same way as in the previous Example, so the generic unfolding has a normal form with the same quadratic terms as (49). We will consider only the η = ±1 cases.

The transformation group (27–28) contributing to the normalization of cubic terms, without introducing quadratic terms, is similar to (50):

(51)  z̃_{1} = a_{1}z_{1} + a_{2}z_{2} + a^{α}w_{α} + p_{1}(z_{1}, z_{2}, w⃗)
z̃_{2} = z_{2} + p_{2}^{1α}w_{1}w_{α} + p_{2}(z_{1}, z_{2}, w⃗)
w̃_{1} = a_{1}w_{1} + P_{1}(w⃗)
w̃_{j} = r_{j}^{α}w_{α} + P_{j}(w⃗), j = 2, …, k

with a_{1} = ±1, a_{2} and a^{α} purely imaginary, and p_{2}^{1α} = 2ia_{1}a^{α}.

If we apply the above transformation in its most general form except that w̃_{j} = w_{j} for j = 2, …, k, and label some terms p_{1}^{α}z_{1}w_{α}, p_{2}^{α}z_{2}w_{α}, the coefficient of e_{2}^{02α}z̄_{1}^{2}t_{α} is transformed in the α = 1 case to

a_{1}e_{2}^{021} + \frac{a_{1}}{2} p_{2}^{1} − \overline{p_{1}^{1}} − ia_{2},

and for α = 2, …, k to

e_{2}^{02α} + \frac{1}{2}p_{2}^{α} − a_{1}\overline{p_{1}^{α}}.

The coefficient of e_{2}^{11α}z_{1}z̄_{1}t_{α} is transformed in the α = 1 case to

a_{1}e_{2}^{111} + a_{1}p_{2}^{1} − (p_{1}^{1} + \overline{p_{1}^{1}}) − 2ia_{2},

and for α = 2, …, k to

e_{2}^{11α} + p_{2}^{α} − a_{1}(p_{1}^{α} + \overline{p_{1}^{α}}).

Unlike the previous Example, the coefficients a^{α} do not contribute to these terms. Even including the imaginary coefficient a_{2}, the transformation group does not have enough degrees of freedom to cancel both the e_{2}^{02α} and e_{2}^{11α} terms simultaneously. We choose to cancel the e_{2}^{11α} coefficients and the imaginary parts of the e_{2}^{02α} coefficients (this is a different choice from the normalization of Proposition 5.8). Then the r_{α}^{β} block can normalize the real coefficients e_{2}^{02α} to (0, 1, 0, …, 0), generically if k ≥ 2. So, the generic k-parameter unfolding of a degenerate parabolic CR singularity has the following cubic normal form:

(52)  z_{2} = z_{1}z̄_{1} + (\frac{1}{2} + t_{2}) · (z_{1}^{2} + z̄_{1}^{2}) + i(z_{1} − z̄_{1})t_{1}
±(z_{1}^{2} + z̄_{1}^{2})z_{1}z̄_{1} + F (z_{1}, z̄_{1}, t) + O(5), k ≥ 2,

where the quartic terms F = F (z_{1}, z̄_{1}, t) depending on t are the same as (33).

For k = 1, or in a degenerate case where the e_{2}^{02α} terms depend only on t_{1}, the real coefficient cannot be re-scaled except by a_{1} = ±1, so there is a real modulus. The remaining cubic coefficients can be normalized exactly as in the previous Example (assuming the nondegeneracy of the quadratic part so there is a i(z_{1} − z̄_{1})t_{1} term), so the generic 1-parameter unfolding (or a non-generic k-parameter unfolding), of a degenerate parabolic CR singularity has the following cubic normal form:

(53)  z_{2} = z_{1}z̄_{1} + (\frac{1}{2} + εt_{1}) · (z_{1}^{2} + z̄_{1}^{2}) + i(z_{1} − z̄_{1})t_{1}
±(z_{1}^{2} + z̄_{1}^{2})z_{1}z̄_{1} + F (z_{1}, z̄_{1}, t) + O(5), ε ≥ 0.

The modulus ε ≥ 0 can be interpreted as a choice of line through the origin in the (t_{1}, t_{2})-plane in the parameter space for (52).

Returning to the normal form (52), the quantity F = F (z_{1}, z̄_{1}, t) can in fact be normalized to zero by a transformation of the form (51). This elimination of F is similar to the result from Example 5.13, but the details of the lengthy calculation are different, and again only sketched here. Let a_{1} = 1, a_{2} = 0 in (51), let p_{1} and p_{2} be as in (34, 35) so they include (among others) terms labeled

p_{1}^{20α}z_{1}^{2}w_{α} + p_{1}^{01α}z_{2}w_{α} + p_{1}^{1αβ}z_{1}w_{α}w_{β} + p_{1}^{K}w^{K},
p_{2}^{11α}z_{1}z_{2}w_{α} + p_{2}^{01αβ}z_{2}w_{α}w_{β},

and let w̃_{1} = w_{1} + P_{1}^{K}w^{K}, w̃_{2} = w_{2} + P_{2}^{αβ}w_{α}w_{β}, w̃_{α} = w_{α}, α = 3, …, k. Then the system of equations from a comparison of coefficients is solvable: p_{1}^{20α} cancels the z̄_{1}^{3}t_{α} terms of F, p_{2}^{11α} cancels the z_{1}z̄_{1}^{2}t_{α} terms, p_{2}^{01αβ} cancels the z̄_{1}^{2}t_{α}t_{β} terms, the real part of p_{1}^{01α} and the imaginary coefficients a^{α} cancel the z_{1}^{2}z̄_{1}t_{α} terms, the imaginary part of p_{1}^{1αβ} and the real coefficients P_{2}^{αβ} cancel the z_{1}z̄_{1}t_{α}t_{β} terms, and the real part of p_{1}^{K} and the real coefficients P_{1}^{K} cancel the z̄_{1}t^{K} terms. The other terms of F do not depend on z̄_{1} and can be canceled by corresponding terms remaining in p_{2}.

In the k = 1 case, F (z_{1}, z̄_{1}, t_{1}) can also be eliminated from (53) (for any ε ≥ 0), but in this case the analogous computation uses the term p_{1}^{2}z_{1}z_{2} from (21) with a real coefficient in place of P_{2}^{αβ} = P_{2}^{11}; the quantity p_{2}(z_{1}, z_{2}) from (21) can compensate for the quartic terms in z_{1}, z̄_{1} this introduces.

Example 5.20. As an example of a one-parameter family of surfaces which is so degenerate that it does not exhibit any of the unfoldings considered in Examples 5.11 – 5.19, but which is interesting from a global point of view as a case where the index sum formula (7) is nonzero, we consider the following map using homogeneous coordinates as a family of embeddings of ℂP^{1} in ℂP^{2}, parametrized by t_{1} ∈ ℝ.

G : ℂP^{1} × ℝ → ℂP^{2}
([z_{0} : z_{1}], t_{1}) ↦ [z_{0} · (z_{0}z̄_{0} + z_{1}z̄_{1}) · (2z_{0}z̄_{0} + z_{1}z̄_{1} − z_{0}z̄_{1} − z̄_{0}z_{1})

: z_{1} · (z_{0}z̄_{0} + z_{1}z̄_{1}) · (2z_{0}z̄_{0} + z_{1}z̄_{1} − z_{0}z̄_{1} − z̄_{0}z_{1})

: t_{1}z_{0} · (2z_{0}z̄_{0} + 3z_{1}z̄_{1}) · (z_{0}z̄_{0} − z_{0}z̄_{1} − z̄_{0}z_{1})].

At t_{1} = 0, this is a holomorphic embedding [z_{0} : z_{1}] ↦ [z_{0} : z_{1} : 0] (so the image G(ℂP^{1} × {0}) has a degenerate CR singularity at every point). For a fixed value of t_{1}, the restriction to one affine neighborhood in the domain ℂP^{1} has image contained in an affine neighborhood of ℂP^{2}:

[1 : z_{1}] ↦ [1 : z_{1} : t_{1} · (2 + 3|z_{1}|^{2}) · (\frac{1}{1 + |z_{1}|^{2}} − \frac{1}{1 + |z_{1} − 1|^{2}} )] .

The image fits inside ℝ^{3} ⊆ ℂ^{2}, and for t_{1} ≠ 0, it has two elliptic points and no hyperbolic points. This is the entire image G(ℂP^{1} × {t_{1}}), except for one point at infinity. A restriction to another pair of affine neighborhoods is:

[z_{0} : 1] ↦ [z_{0} : 1 : \frac{t_{1}z_{0} · (2z_{0}z̄_{0} + 3) · (z_{0}z̄_{0} − z_{0} − z̄_{0})}{(z_{0}z̄_{0} + 1) · (2z_{0}z̄_{0} + 1 − z_{0} − z̄_{0})}] ,

which for t_{1} ≠ 0 is in standard position (8), but not flattened inside ℝ^{3}. Observing the z_{0}z̄_{0} term in the numerator, the view in this neighborhood shows a third elliptic point at [0 : 1 : 0]. This example is consistent with the calculation of [F] §7 (see also [IS] App. IV), showing that there is a small perturbation of a complex projective line in ℂP^{2} into a real surface in general position, with index sum ∑ ind(x) = ∫ c_{1}(F, J) = 3 (where (F, J) is the restriction of T ℂP^{2} to the surface), and exactly three positively oriented elliptic points.

5.3. Visualization.

From the global theory of immersed compact surfaces discussed in Section 2, one expects CR singularities disappearing (or appearing) in a deformation will generally cancel (or be created) in pairs: one elliptic and one hyperbolic, with the same orientation. Example 5.18 represents a local version of this phenomenon, and the normal form (49) for the unfolding flattens (at least up to +O(3)) to fit in ℝ^{3} × ℝ, so it can be illustrated by a series of pictures.

In fact, for most of the examples of unfoldings ; M in the previous Subsection, we were able to choose normal forms so that for each t, the slice M_{t} is contained in ℝ^{3} × {t}, at least up to some degree — there may be higher degree terms with nonzero imaginary parts. Here, we truncate the defining equations to their lower-degree, holomorphically flattened normal forms, to view a graphical representation of the unfolding of M ⊆ ℝ^{3} × {0}. In the following Figures, we visualize ℝ^{3} ⊆ ℂ^{2} as the x_{1}, y_{1}, x_{2}-subspace, where the x_{1}, y_{1}-plane is the z_{1}-axis and the horizontal planes parallel to it are also complex lines. Each of the normal forms for surfaces M in Proposition 5.1 is in standard position, so that the CR singularity is at the origin, and the x_{1}, y_{1}-plane is the complex tangent line. For some unfoldings ; M, a slice M_{t} could have a CR singularity at some point other than the origin of ℂ^{2} × {t}, or could have a complex tangent line other than the z_{1}-axis, although when M_{t} fits inside ℝ^{3} × {t}, the only possible complex tangent lines are horizontal planes parallel to the z_{1}-axis. So, in these pictures rendered by [POV-Ray], the CR singularities will be visible as horizontal tangents, i.e., the familiar critical points of the real height function in the x_{2} direction.

Example 5.21. In Figure 1, the elliptic CR singularity of a surface

z_{2} = z_{1}z̄_{1} + γ(z_{1}^{2} + z̄_{1}^{2}), 0 < γ < \frac{1}{2}

is shown as the vertex of an elliptic paraboloid, tangent to the z_{1}-axis at the origin. It meets complex lines (horizontal planes) in ellipses, or in a single point, at the CR singularity.

Example 5.22. In Figure 2, the hyperbolic CR singularity of a surface

z_{2} = z_{1}z̄_{1} + γ(z_{1}^{2} + z̄_{1}^{2}), \frac{1}{2} < γ < ∞

is the saddle point, located at the crossing-point of the ×-shaped degenerate conic level set. The other level curves are hyperbolas. The Figure 1. Elliptic CR singularity

Figure 2. Hyperbolic CR singularity

angle at the singular point, formed by the intersection of the surface with the complex tangent line, is clearly a biholomorphic invariant.

Looking at the quadric models for generic elliptic and hyperbolic singularities, their non-trivial intersections with complex lines parallel to the x_{1}, y_{1}-plane are conics of a constant eccentricity. Varying the parameter t_{1} near 0 in the generic unfolding (31) varies the eccentricity, and the angle in the hyperbolic case.

Figure 3. Nondegenerate parabolic CR singularity

Example 5.23. In Figure 3, the nondegenerate parabolic CR singularity of the surface

z_{2} = z_{1}z̄_{1} + \frac{1}{2}(z_{1}^{2} + z̄_{1}^{2}) + i(z_{1} − z̄_{1})z_{1}z̄_{1}

is located at the cusp of the ≻-shaped singular level set.

A nondegenerate unfolding ; M of the above parabolic singularity, as in Example 5.18 with k = 1, is the cubic normal form:

(54) z_{2} = z_{1}z̄_{1} + \frac{1}{2}(z_{1}^{2} + z̄_{1}^{2}) + i(z_{1} − z̄_{1})t_{1} + i(z_{1} − z̄_{1})z_{1}z̄_{1}.

For t = (t_{1}) with t_{1} < 0, there is a pair of CR singularities in the deformed surface M_{t}, one elliptic and one hyperbolic, as shown in Figure 4, and for t_{1} > 0, the surface M_{t} is totally real in ℂ^{2} × {t}, showing no horizontal tangents in Figure 5. The unfolding in Figures 3–5 is similar to Figure 1 of [Callahan], illustrating graphs of a family of smooth functions undergoing a “catastrophe” as the number of local minimum points varies with the parameter t.

Figure 4. Deformation, t_{1} < 0, of nondegenerate parabolic CR singularity into an elliptic/hyperbolic pair of CR singularities

Figure 5. Totally real deformation, t_{1} > 0, of nondegenerate parabolic CR singularity

To more precisely analyze the local geometry of the unfolding, we return to the complexification construction of Section 4. Corresponding to the smooth real variety ; M ⊆ ℂ^{2+1} is a smooth complex variety ; M_{c} ⊆ ℂ^{2(2+1)}, parametrized by Σ : ℂ^{2+1} → ℂ^{2(2+1)}. Let ℂ^{2+1} have coordinates (z_{1}, ζ_{1}, ω_{1}). The image of ; M_{c} under the projection π : ℂ^{2(2+1)} → ℂ^{2+1} is all of ℂ^{2+1}; the map π ∘ Σ as in (19):

π ∘ Σ : ℂ^{2+1} → ℂ^{2+1} : (z_{1}, ζ_{1}, ω_{1})
↦ (z_{1}, z_{1}ζ_{1} + \frac{1}{2}(z_{1}^{2} + ζ_{1}^{2}) + i(z_{1} − ζ_{1})ω_{1} + i(z_{1} − ζ_{1})z_{1}ζ_{1}, ω_{1})^{T}

is a two-to-one branched cover near (0, 0). At each point in the domain ℂ^{2+1}, the complex Jacobian of this polynomial map either will have full rank, 3, or will be singular, with rank 2. The singular locus in the domain is the complex affine quadric variety

{z_{1} + ζ_{1} − iω_{1} + iz_{1}^{2} − 2iz_{1}ζ_{1} = 0},

and, near the origin, this is also exactly the locus where π ∘ Σ is oneto-one.

As in (18), let δ denote the inclusion of the totally real subspace

\{ζ_{1} = z̄_{1}, ω_{1} = \overline{ω_{1}} = t_{1}\}

in ℂ^{2+1}, so ; M is the image of π ∘ Σ ∘ δ. The intersection of this real subspace with the singular locus is

{(x_{1} + iy_{1}, x_{1} − iy_{1}, t_{1}) : 2x_{1} − 2x_{1}y_{1} = 0, t_{1} + x_{1}^{2} + 3y_{1}^{2} = 0}.

The first condition factors as 2x_{1}(1 − y_{1}) = 0, and if we are only considering points in ; M near (0, 0), the solution set is {x_{1} = 0, y_{1} = ±\sqrt{−t_{1}/3}\}. So, the candidates for CR singular points in ; M near (0, 0) are of the form

\{(z_{1}, z_{2}, t_{1})^{T} = (±i\sqrt{−t_{1}/3}, ±4(−t_{1}/3)^{3/2}, t_{1})^{T} \} ⊆ ; M ⊆ ℂ^{2+1},

or in implicit form, this locus is the real twisted cubic curve {t_{1} +3y_{1}^{2} = 0, x_{2} − 4y_{1}^{3} = 0} in the y_{1}, x_{2}, t_{1} real coordinate subspace, tangent to the y_{1}-axis.

In fact, this real curve is the CR singular locus N_{1} of ; M near (0, 0), as can be seen by fixing t = (t_{1}) with t_{1} < 0, and moving the slice M_{t} into standard position in ℂ^{2} × {t}. Corresponding to one candidate point in the slice M_{t}, we first consider a translation of the form z̃_{1} = z_{1} − i\sqrt{−t_{1}/3}. Substituting z̃_{1} + i\sqrt{−t_{1}/3} for z_{1} in RHS of (54) (and dropping the tilde) gives:

4(−\frac{t_{1}}{3} )^{\frac{3}{2}} + (1 − 4\sqrt{−\frac{t_{1}}{3}} )z_{1}z̄_{1} + (\frac{1}{2} + \sqrt{−\frac{t_{1}}{3}} )(z_{1}^{2} + z̄_{1}^{2}) + i(z_{1} − z̄_{1})z_{1}z̄_{1},

so translating and then rescaling z_{2} gives the normal form:

z_{2} = z_{1}z̄_{1} + γ_{t}(z_{1}^{2} + z̄_{1}^{2}) + O(3),

where γ_{t} = \frac{\frac{1}{2} +\sqrt{−t_{1}/3}}{1−4\sqrt{−t_{1}/3}}. It can be concluded that the tangent plane

to M_{t} at the original candidate point is a complex line parallel to the z_{1}-axis (as expected from Figure 4), and the CR singularity is hyperbolic for \frac{−3}{16} < t_{1} < 0. The other substitution, z̃_{1} − i\sqrt{−t_{1}/3},

Figure 6. Degenerate parabolic CR singularity, η = −1

translating the other candidate point, leads to a similar calculation

but with γ_{t} = \frac{\frac{1}{2} −\sqrt{−t_{1}/3}}{1+4\sqrt{−t_{1}/3}}, at the elliptic CR singularity.

The property that the complexified parametrization π ∘ Σ is a twoto-one ramified map (locally, near the parabolic point) is shared with the generic elliptic and hyperbolic points of real n-manifolds in ℂ^{n}, as considered by [MW]. The observation that the curve N_{1} is tangent to the complex z_{1}-axis is an example of the intrinsic characterization of the parabolic point of the real 3-manifold ; M in ℂ^{3}, as in Remark 5.9.

Example 5.24. A surface with a degenerate parabolic CR singularity has a quartic normal form, as in Proposition 5.1:

(55) z_{2} = z_{1}z̄_{1} + \frac{1}{2}(z_{1}^{2} + z̄_{1}^{2}) + η(z_{1}^{2} + z̄_{1}^{2})z_{1}z̄_{1} + O(5), η = +1, −1, 0.

We will consider the η = ±1 cases only, and drop the O(5) part in the following illustrations.

In the η = −1 case, (55) simplifies to

x_{2} = 2x_{1}^{2} − 2x_{1}^{4} + 2y_{1}^{4},

the graph of which is a surface M^{−} with a local minimum at the origin in ℝ^{3} (the y_{2} = 0 subspace) in Figure 6.

The critical point of the height function is degenerate in the sense of Morse Theory, being flat in the y_{1} direction.

Figure 7. Degenerate parabolic CR singularity, η = +1

In the η = +1 case, (55) simplifies to

x_{2} = 2x_{1}^{2} + 2x_{1}^{4} − 2y_{1}^{4},

a surface M^{+} which has a degenerate saddle point at the origin in ℝ^{3} in Figure 7.

The level set of a nondegenerate saddle point has a × shape as in Figure 2, but the level set intersecting the critical point of the height function in Figure 7 is a self-tangent curve. The level sets of Figures 1–3, 6, and 7 resemble the sketches in [Martinet] §3.XII.5, of level curves of smooth functions of two real variables with critical points.

One expects from Morse Theory that smooth perturbations of the surface M^{+} or M^{−} in ℝ^{3} will replace one of these degenerate critical points by one or more nondegenerate critical points. The normal form calculation (52) from Example 5.19 shows that a generic 2-parameter unfolding of M^{±} in ℂ^{2} is u-equivalent (up to fourth degree, continuing to neglect O(5)) to the following ; M^{±} ⊆ ℝ^{3} ×ℝ^{2} ⊆ ℂ^{2} ×ℝ^{2}, where each slice M_{t}^{±} is visible in ℝ^{3} × {t}:

(56)z_{2} = z_{1}z̄_{1} + (\frac{1}{2} + t_{2}) · (z_{1}^{2} + z̄_{1}^{2}) + i(z_{1} − z̄_{1})t_{1} ± (z_{1}^{2} + z̄_{1}^{2})z_{1}z̄_{1}

Figure 8. Deformation of the η = −1 degenerate parabolic CR singularity, t_{1} ≠ 0, t_{2} = 0

Figure 9. Deformation of the η = −1 degenerate parabolic CR singularity, t_{1} = 0, t_{2} < 0

In the η = −1 case, varying the first unfolding parameter t_{1} while fixing the second, t_{2} = 0, introduces a linear (in y_{1}) term in the real defining equation:

x_{2} = 2x_{1}^{2} − 2x_{1}^{4} + 2y_{1}^{4} − 2t_{1}y_{1}.

The slices M_{t}^{−} = M_{(t_{1},0)}^{−} have a nondegenerate elliptic point away from the origin, as in Figure 8.

Continuing with η = −1, fixing the first unfolding parameter t_{1} = 0 while choosing nonzero values for the second, t_{2}, introduces a quadratic (in x_{1}, y_{1}) term in the real defining equation:

x_{2} = 2x_{1}^{2} − 2x_{1}^{4} + 2y_{1}^{4} + 2t_{2}(x_{1}^{2} − y_{1}^{2}).

For t_{2} < 0, the slices M_{t}^{−} = M_{(0,t_{2})}^{−} have a nondegenerate elliptic point at the origin, as in Figure 9, but for t_{2} > 0, there is a hyperbolic CR singularity at the origin and there are two elliptic points nearby, as in Figure 10.

Figure 10. Deformation of the η = −1 degenerate parabolic CR singularity, t_{1} = 0, t_{2} > 0

Figure 11. Deformation of the η = +1 degenerate parabolic CR singularity, t_{1} ≠ 0, t_{2} = 0

In the η = +1 case, varying the first unfolding parameter t_{1} while fixing the second, t_{2} = 0, gives slices M_{t}^{+} = M_{(t_{1},0)}^{+} with a nondegenerate hyperbolic point away from the origin, as in Figure 11.

Figure 12. Deformation of the η = +1 degenerate parabolic CR singularity, t_{1} = 0, t_{2} > 0

Figure 13. Deformation of the η = +1 degenerate parabolic CR singularity, t_{1} = 0, t_{2} < 0

Continuing with η = +1, fixing the first unfolding parameter t_{1} = 0 while varying the second, t_{2}, gives slices M_{t}^{+} = M_{(0,t_{2})}^{+} with one nondegenerate hyperbolic point at the origin for t_{2} > 0, as in Figure 12, but for t_{2} < 0, there is an elliptic CR singularity at the origin and two hyperbolic points nearby, as in Figure 13.

All these unfoldings of the degenerate and nondegenerate parabolic points are consistent with the conservation of topological index sum of real surfaces in ℂ^{2} under perturbations as in Section 2. The unfolding of the nondegenerate parabolic point has a pair of points with indices +1 and −1 before they cancel and the surface becomes totally real. The slices in the unfolding of the degenerate parabolic case with η = −1 have index sum +1 near the origin, when there are 1 or 3 nondegenerate critical points; and similarly, the slices with nondegenerate singularities in the η = +1 case have a local sum of −1.

To more precisely analyze the local geometry of the unfoldings ; M^{±}, we turn again to the complexification construction of Section 4. Corresponding to the smooth real variety ; M^{+} ⊆ ℂ^{2+2} is a smooth complex variety ; M_{c}^{+} ⊆ ℂ^{2(2+2)}, parametrized by Σ : ℂ^{2+2} → ℂ^{2(2+2)}. Let ℂ^{2+2} have coordinates (z_{1}, ζ_{1}, ω_{1}, ω_{2}). The image of ; M_{c}^{+} under the projection π : ℂ^{2(2+2)} → ℂ^{2+2} is all of ℂ^{2+2}; the map π ∘ Σ as in (19):

π ∘ Σ : ℂ^{2+2} → ℂ^{2+2} : (z_{1}, ζ_{1}, ω_{1}, ω_{2}) ↦

(z_{1}, z_{1}ζ_{1} + (\frac{1}{2} + ω_{2}) · (z_{1}^{2} + ζ_{1}^{2}) + i(z_{1} − ζ_{1})ω_{1} + (z_{1}^{2} + ζ_{1}^{2})z_{1}ζ_{1}, ω_{1}, ω_{2})^{T}

is a two-to-one branched cover near (0, 0). (Globally, points in the target can have up to three inverse images, but points near the origin have at most two inverse images near the origin in the domain.) At each point in the domain ℂ^{2+2}, the complex Jacobian of this polynomial map either will have full rank, 4, or will be singular, with rank 3. The singular locus in the domain is the complex affine variety

\{z_{1} + (\frac{1}{2} + ω_{2})2ζ_{1} − iω_{1} + z_{1}^{3} + 3z_{1}ζ_{1}^{2} = 0\}.

As in (18), let δ denote the inclusion of the totally real subspace

\{ζ_{1} = z̄_{1}, ω_{1} = \overline{ω_{1}} = t_{1}, ω_{2} = \overline{ω_{2}} = t_{2}\}

in ℂ^{2+2}, so ; M^{+} is the image of π ∘ Σ ∘ δ. The intersection of this real subspace with the singular locus is

{(x_{1} + iy_{1}, x_{1} − iy_{1}, t_{1}, t_{2}) : 2(1 + t_{2})x_{1} + 4x_{1}^{3} = 0, 4y_{1}^{3} + 2t_{2}y_{1} + t_{1} = 0}.

The first condition factors as 2x_{1}(1 + t_{2} + 2x_{1}^{2}) = 0, and if we are only considering points in ; M^{+} near (0, 0), the solution set in the x_{1}, y_{1}-plane for each t = (t_{1}, t_{2}) is a set of at most three points on the y_{1}-axis, as in Figures 7, 11–13. So, the candidates for CR singular points in ; M near (0, 0) are of the form

{(z_{1}, z_{2}, t_{1}, t_{2})^{T} = (iy_{1}, −2t_{2}y_{1}^{2} − 2t_{1}y_{1} − 2y_{1}^{4}, t_{1}, t_{2})^{T} } ⊆ ; M^{+} ⊆ ℂ^{2+2},

Figure 14. Discriminant locus in the t_{1}, t_{2} parameter space for the unfolding ; M^{+}

where y_{1} satisfies 4y_{1}^{3} + 2t_{2}y_{1} + t_{1} = 0. This locus is a surface contained in the real y_{1}, x_{2}, t_{1}, t_{2} subspace, which is smooth near the origin, and its tangent plane at the origin is the y_{1}, t_{2}-plane.

The discriminant of the cubic equation 4y_{1}^{3} + 2t_{2}y_{1} + t_{1} = 0 is −432t_{1}^{2} − 128t_{2}^{3}, and the zero locus of this curve is plotted in the t_{1}, t_{2} parameter space in Figure 14.

When t_{1} = t_{2} = 0, the cubic equation has a triple root, and the slice M_{0}^{+} has an isolated degenerate parabolic CR singularity as in Figure 7. When t = (t_{1}, t_{2}) is a point above the ⋏-shaped curve, so t_{2} > −\frac{3}{2}t_{1}^{2/3}, the cubic has one real root and M_{t}^{+} has an isolated hyperbolic CR singularity at (z_{1}, z_{2})^{T} = (iy_{1}, −2t_{2}y_{1}^{2} − 2t_{1}y_{1} − 2y_{1}^{4})^{T} as in Figures 11 and 12. When t is below the curve, the cubic equation has three distinct roots and M_{t}^{+} has two hyperbolic CR singularities and one elliptic CR singularity near 0. For points t = (t_{1}, −\frac{3}{2}t_{1}^{2/3}) on the curve, but not at the cusp at the origin, the cubic has two solutions, a simple root at y_{1} = −t_{1}^{1/3}, and a double root at y_{1} = \frac{1}{2}t_{1}^{1/3}. Near 0, M_{t}^{+} has a hyperbolic CR singularity at (z_{1}, z_{2})^{T} = (−it_{1}^{1/3}, 3t_{1}^{4/3})^{T}, and a nondegenerate parabolic CR singularity at (z_{1}, z_{2})^{T} = (\frac{i}{2}t_{1}^{1/3}, −\frac{3}{8}t_{1}^{4/3})^{T}. This can be verified by a translation to standard position and then transformation to normal form as in the previous Example. In this case, the hyperbolic singularity of M_{t}^{+} has γ_{t} = \frac{1}{2} · \frac{1+3t_{1}^{2/3}}{1−6t_{1}^{2/3}} , > \frac{1}{2} for t_{1}

close to but not equal to 0. The parabolic point of M_{t}^{+} can be put, by a holomorphic (but nonlinear) transformation, into the form

z_{2} = z_{1}z̄_{1} + \frac{1}{2}(z_{1}^{2} + z̄_{1}^{2}) − \frac{4it_{1}^{1/3}}{3t_{1}^{2/3} − 2}(z_{1} − z̄_{1})z_{1}z̄_{1} + O(4),

so it is a nondegenerate parabolic point for t_{1} close to but not equal to 0, as in Proposition 5.1.

Figure 14 resembles Figure 4 of [Callahan], and the overall geometry of this unfolding resembles the analogous catastrophe phenomenon described by [Callahan]. Considering ; M^{+} as a four-dimensional submanifold of ℂ^{4}, it has a degenerate parabolic CR singularity at the origin, in the sense of Proposition 5.8. The CR singular locus N_{1} is a totally real surface in ; M^{+}, and its tangent plane at the origin meets the complex tangent line in the y_{1}-axis, the parabolic line as in Remark 5.9. The locus of parabolic points

\{(z_{1}, z_{2}, w_{1}, w_{2})^{T} = (\frac{i}{2}t_{1}^{1/3}, −\frac{3}{8}t_{1}^{4/3}, t_{1}, −\frac{3}{2}t_{1}^{2/3})^{T} \}

is a smooth curve, which could be re-parametrized as:

(\frac{i}{2}y_{1}, −\frac{3}{8}y_{1}^{4}, y_{1}^{3}, −\frac{3}{2}y_{1}^{2})^{T} ,

also tangent to the y_{1}-axis. The tangency of the parabolic locus to the parabolic line is an intrinsic characterization of the degenerate parabolic point of an n-manifold in ℂ^{n} (Proposition 4.1 of [Webster_{2}]).

The geometry of the unfolding ; M^{−} is analyzed by similar calculations. For t = (t_{1}, t_{2}), the z_{1} coordinates of the CR singularities of the slice M_{t} are iy_{1}, where y_{1} is a solution of the cubic equation −4y_{1}^{3} + 2t_{2}y_{1} + t_{1} = 0. The discriminant is 128t_{2}^{3} − 432t_{1}^{2}, so its zero locus is just an upside-down Figure 14. The slice M_{t} will have two elliptic points and one hyperbolic point as in Figure 10 for t above the ⋎-shaped curve, one elliptic point as in Figures 8, 9, for t below the curve, and one elliptic point and one nondegenerate parabolic point for t ≠ 0 on the curve.

6. Real m-submanifolds in ℂ^{n}, m < n

We move to the case of higher dimension, or actually higher codimension, where m < n. In some of the results of this Section, we will also restrict our attention to the case m = \frac{2}{3}(n+1), where M in general position will be totally real except for isolated points in N_{1}.

6.1. Normal forms.

We recall from [Beloshapka] and [C_{1}], without re-working all of the details, the derivation of a normal form and the nondegeneracy conditions for the defining equations of M, in the stable dimension range \frac{2}{3}(n + 1) ≤ m < n. The equations in standard position as in (8) are:

H_{σ}(z_{1}, z̄_{1}, x) = α_{σ}z_{1}^{2} + β_{σ}z_{1}z̄_{1} + γ_{σ}z̄_{1}^{2} + (∑ δ_{σ}^{σ_{1}}z_{1}x_{σ_{1}})

+ (∑ ϵ_{σ}^{σ_{1}} z̄_{1}x_{σ_{1}}) + (∑ θ_{σ}^{σ_{1}σ_{2}}x_{σ_{1}}x_{σ_{2}}) + E_{σ}(z_{1}, z̄_{1}, x) (57)

h_{u}(z_{1}, z̄_{1}, x) = α_{u}z_{1}^{2} + β_{u}z_{1}z̄_{1} + γ_{u}z̄_{1}^{2} + (∑ δ_{u}^{σ_{1}}z_{1}x_{σ_{1}})
+ (∑ ϵ_{u}^{σ_{1}} z̄_{1}x_{σ_{1}}) + (∑ θ_{u}^{σ_{1}σ_{2}}x_{σ_{1}}x_{σ_{2}}) + e_{u}(z_{1}, z̄_{1}, x),

where E_{σ} = O(3) is real valued for σ = 2, …, m − 1, and e_{u} = O(3) is complex valued for u = m, …, n.

The \underline{first nondegeneracy condition} is that the (n − m + 1) × 2 block of coefficients β_{u}, γ_{u} in the functions h_{u} satisfies:

(58)  rank ( \begin{matrix}β_{m} & γ_{m} \\ ⋮ & ⋮ \\ β_{n} & γ_{n} \end{matrix} ) = 2.

In the nondegenerate case, there is a holomorphic transformation in a neighborhood of 0 in ℂ^{n} taking M to the following partial normal form:

(59) y_{σ} = H_{σ}(z_{1}, z̄_{1}, x) = 0 + E_{σ}(z_{1}, z̄_{1}, x) = O(3)
(60) z_{τ} = h_{τ}(z_{1}, z̄_{1}, x) = (∑ ϵ_{τ}^{σ_{1}} z̄_{1}x_{σ_{1}}) + e_{τ}(z_{1}, z̄_{1}, x)
z_{n−1} = h_{n−1}(z_{1}, z̄_{1}, x) = z̄_{1}^{2} + e_{n−1}(z_{1}, z̄_{1}, x)
(61) z_{n} = h_{n}(z_{1}, z̄_{1}, x) = z_{1}z̄_{1} + (∑ ϵ_{n}^{σ_{1}} z̄_{1}x_{σ_{1}}) + e_{n}(z_{1}, z̄_{1}, x),

for τ = m …, n − 2, or there are no h_{τ} expressions if m = n − 1. (The quantities E_{σ}, e_{u} may have changed but are still O(3).)

6. REAL m-SUBMANIFOLDS IN ℂ^{n}, m < n 49

The real and imaginary parts of the coefficients ϵ_{u}^{σ_{1}} in (60) and (61), for u = m, …, n − 2 and u = n, on the terms z̄_{1}x_{σ_{1}}, σ_{1} = 2, …, m − 1, form a real 2(n − m) × (m − 2) coefficient matrix, in this expression where the LHS is a column (n − m)-vector:

(∑ ϵ_{u}^{σ_{1}} z̄_{1}x_{σ_{1}})_{u=m,...,n−2,n} = (62)

( \begin{matrix}1 & i … 0 & 0 \\ ⋮ & & ⋮ \\ 0 & 0 … 1 & i \end{matrix} )(\begin{matrix}Re(ϵ_{m}^{2}) & Re(ϵ_{m}^{3}) … & Re(ϵ_{m}^{m−1}) \\ Im(ϵ_{m}^{2}) & Im(ϵ_{m}^{3}) … & Im(ϵ_{m}^{m−1}) \\ ⋮ & & ⋮ \\ Re(ϵ_{n−2}^{2}) & Re(ϵ_{n−2}^{3}) … & Re(ϵ_{n−2}^{m−1}) \\ Im(ϵ_{n−2}^{2}) & Im(ϵ_{n−2}^{3}) … & Im(ϵ_{n−2}^{m−1}) \\ Re(ϵ_{n}^{2}) & Re(ϵ_{n}^{3}) … & Re(ϵ_{n}^{m−1}) \\ Im(ϵ_{n}^{2}) & Im(ϵ_{n}^{3}) … & Im(ϵ_{n}^{m−1}) \end{matrix})( x_{2} ; ⋮ ; x_{m−1} )z̄_{1}.

The \underline{second nondegeneracy condition} is that this matrix has rank 2(n − m). In the nondegenerate case, there is a linear transformation (the R block from the matrix A from (9)) taking this coefficient matrix to an echelon form, so the h_{τ} expressions (60), if any, become

(63)  (x_{2(τ−m+2)} + ix_{2(τ−m+2)+1})z̄_{1} + e_{τ}(z_{1}, z̄_{1}, x),

and the h_{n} expression (61) becomes

(64)  z_{1}z̄_{1} + x_{2}z̄_{1} + ix_{3}z̄_{1} + e_{n}(z_{1}, z̄_{1}, x).

Proposition 6.1 ([C_{6}]). Given \frac{2}{3}(n + 1) ≤ m < n, if M is a real analytic m-submanifold of ℂ^{n} with a CR singularity satisfying both nondegeneracy conditions, then there exists a holomorphic coordinate change z̃ = Az⃗+p⃗ as in (9), in a neighborhood of 0 ∈ ℂ^{n}, transforming the equations (59–61) into the following real algebraic normal form:

(65) ỹ_{σ} = 0, σ = 2, …, m − 1
z̃_{τ} = (x̃_{2(τ−m+2)} + ix̃_{2(τ−m+2)+1})¯z̃_{1}, τ = m …, n − 2
z̃_{n−1} = ¯ z̃_{1}^{2}
z̃_{n} = (z̃_{1} + x̃_{2} + ix̃_{3})¯z̃_{1}.

So, M can be transformed into the real algebraic variety (65), denoted ; M, with the higher degree terms E_{σ}, e_{u} eliminated entirely in a neighborhood of the CR singularity, assuming only that M is real analytic and satisfies both second order nondegeneracy conditions at a point in N_{1}.

One could flatten ; M to fit inside ℝ^{n+1} ⊆ ℂ^{n}, by introducing some new quadratic terms to get

h_{τ}(z_{1}, z̄_{1}, x) = (z̄_{1} + z_{1})x_{2(τ−m+2)} + i(z̄_{1} − z_{1})x_{2(τ−m+2)+1}
h_{n−1}(z_{1}, z̄_{1}, x) = z_{1}^{2} + z̄_{1}^{2}
h_{n}(z_{1}, z̄_{1}, x) = z_{1}z̄_{1} + (z̄_{1} + z_{1})x_{2} + i(z̄_{1} − z_{1})x_{3},

but this does not help as much with the visualization as the flattened normal forms from the previous Section.

So, M has the property of “algebraizability”: having a real algebraic representative in its equivalence class. Some real analytic surfaces in ℂ^{2} also have this property (Examples 5.4, 5.13), but unlike the case of surfaces in ℂ^{2}, for M as in Proposition 6.1, there is no continuous invariant. As remarked in [C_{6}], the algebraic normal form (65) is analogous to, but different from, the simplest type of singularity for differentiable maps ℂ^{m} → ℂ^{n}, where the components of the parametric map in normal form are monomials.

In the case m = \frac{2}{3}(n + 1), the CR singular point at 0 is an isolated point and the real coefficient matrix (62) is square, with size 2(n − m) × 2(n − m). If the linear transformation A is required to preserve the orientation of T0M, then the nondegenerate normal form

h_{n} = z_{1}z̄_{1} + x_{2}z̄_{1} − ix_{3}z̄_{1} + e_{n}(z_{1}, z̄_{1}, x)

is inequivalent to the formula (64) with the other sign. They are related by the holomorphic transformation z̃_{3} = −z_{3}, but this switches the orientation of T0M. Recalling the topological description of CR singularities from Section 2, these two nondegenerate normal forms correspond to the differential-topological intersection index, +1 or −1, where the Gauss map M → G meets the oriented submanifold D_{1} \ D_{2} ⊆ G.

Continuing under the assumption m = \frac{2}{3}(n + 1), if the first nondegeneracy condition is satisfied but the second is not, then the real coefficient matrix in (62) has less than full rank. Assuming it has rank 2(n − m) − 1, a real linear transformation of the x_{2}, …, x_{m−1} coordinates that puts the real matrix into echelon form can make the second column (corresponding to x_{3}) of the matrix zero, so the expressions (63) are the same but expression (61) becomes

h_{n} = z_{1}z̄_{1} + ϵ_{n}^{2}z̄_{1}x_{2} + (∑_{ℓ=4}^{m−1} ϵ_{n}^{ℓ}z̄_{1}x_{ℓ}) + O(3),

where ϵ_{n}^{2} is a nonzero complex coefficient and for ℓ = 4, …, m − 1, ϵ_{n}^{ℓ} is purely imaginary. A complex linear transformation of the form

z̃ = (a_{1}z_{1}, z_{2}, …, z_{m−1}, \overline{a_{1}}z_{m}, …, \overline{a_{1}}z_{n−2}, \overline{a_{1}}^{2}z_{n−1}, |a_{1}|^{2}z_{n})^{T}

can normalize ϵ_{n}^{2} to 1, transforming h_{n} into:

h_{n} = z_{1}z̄_{1} + z̄_{1}x_{2} + (∑_{ℓ=4}^{m−1} ϵ_{n}^{ℓ}z̄_{1}x_{ℓ}) + O(3),

where now the coefficients ϵ_{n}^{ℓ} are complex (using the same symbol even though the value of the coefficients may have changed). In the (m, n) = (4, 5) case, there are no terms besides z_{1}z̄_{1} + z̄_{1}x_{2}. Otherwise, the remaining terms satisfy the identity:

(66) ∑_{ℓ=4}^{m−1} ϵ_{n}^{ℓ}z̄_{1}x_{ℓ} = ∑_{j=2}^{\frac{m}{2} −1}(ϵ_{n}^{2j}z̄_{1}x_{2j} + ϵ_{n}^{2j+1}z̄_{1}x_{2j+1})

= ∑_{j=2}^{\frac{m}{2} −1} ((Im(ϵ_{n}^{2j+1}) + iIm(ϵ_{n}^{2j})) · (x_{2j} + ix_{2j+1})z̄_{1}
+(Re(ϵ_{n}^{2j}) − Im(ϵ_{n}^{2j+1}))z̄_{1}x_{2j}
+(Re(ϵ_{n}^{2j+1}) + Im(ϵ_{n}^{2j}))z̄_{1}x_{2j+1}) .

The re-grouping allows the elimination of the (x_{2j} + ix_{2j+1})z̄_{1} terms by

linear transformations of the form z̃_{n} = z_{n} + ∑_{τ=m}^{n−2} a_{n}^{τ}z_{τ}, and the remain-

ing z̄_{1}x_{ℓ} terms have real coefficients so they can be eliminated by a real

linear transformation of the x_{2} variable, z̃_{2} = z_{2} + ∑_{ℓ=4}^{m−1} r_{2}^{ℓ}z_{ℓ}.

Note that it was at this point where we used the assumption m = \frac{2}{3}(n + 1). For \frac{2}{3}(n + 1) < m < n, there would be terms z̄_{1}x_{ℓ}, with ℓ = 4, …, 2n − 2m + 1, 2n − 2m + 2, …, m − 1, and only enough z_{τ} quantities available to cancel ℓ = 4, …, 2n − 2m + 1.

This leaves the following quadratic normal form for the defining equations of M near the origin:

(67)  y_{σ} = 0 + E_{σ}(z_{1}, z̄_{1}, x) = O(3)
z_{τ} = (x_{2(τ−m+2)} + ix_{2(τ−m+2)+1})z̄_{1} + e_{τ}(z_{1}, z̄_{1}, x)
z_{n−1} = z̄_{1}^{2} + e_{n−1}(z_{1}, z̄_{1}, x),
z_{n} = z_{1}z̄_{1} + z̄_{1}x_{2} + e_{n}(z_{1}, z̄_{1}, x).

We will not consider degenerate CR singularities where the real coefficient matrix has rank less than 2(n − m) − 1, or where the first nondegeneracy condition fails. The remainder of this Section will deal with only the nondegenerate normal form (65) and the degenerate normal form (67).

Writing out all the cubic terms in m variables in the defining equations (57) would give a lengthy expression. To get a normal form for (67), using the transformation group (9) to eliminate as many terms as possible to simplify the expression, we take three shortcuts.

First, any cubic term in (57) not depending on z̄_{1}, that is, having only z_{1}, x_{2}, …, x_{m−1} factors, can be eliminated by a holomorphic transformation of the form z̃_{j} = z_{j} + p_{j}(z_{1}, z_{2}, …, z_{m−1}), j = 2, …, n, which does not change any other cubic terms in the system of equations, with the possible exception of introducing a complex conjugate term (involving z̄_{1} and not z_{1}) in the H_{σ} expressions.

Second, we can take advantage of the fact that h_{n−1} and h_{n} expressions in the normal form (67) have quadratic part identical (up to re-numbered subscripts) to that of the nondegenerate normal form for real threefolds in ℂ^{4}, as in [C_{5}]. It was shown there that all cubic terms of h_{n−1}, h_{n} in z_{1}, z̄_{1}, x_{2} can be eliminated by a holomorphic transformation z̃ = z⃗ + p⃗(z_{1}, z_{2}, z_{n−1}, z_{n}).

Third, the E_{σ}, σ = 4, …, m − 1 and

h_{τ} = (x_{2(τ−m+2)} + ix_{2(τ−m+2)+1})z̄_{1} + O(3)

quantities have no analogue in [C_{5}], but are identical to the expressions in [C_{6}]. It was shown in the Proofs of Theorem 5.6 and Corollary 5.7 of [C_{6}] that for each τ = m, …, n − 2, all of the cubic terms of the three expressions h_{τ}, E_{2(τ−m+2)}, and E_{2(τ−m+2)+1} can be eliminated by a holomorphic transformation of the form z̃_{τ} = z_{τ} + p_{τ}, z̃_{2(τ−m+2)} = z_{2(τ−m+2)} + p_{2(τ−m+2)}, z̃_{2(τ−m+2)+1} = z_{2(τ−m+2)+1} + p_{2(τ−m+2)+1}, without contributing cubic terms to any of the other equations. An analogous argument applies in this case: the same calculation solving the system of three equations works, the only minor difference being in the quadratic part of h_{n}. It also follows from an analogy with the treatment of the E_{6} equation from the Proof of Theorem 5.6 of [C_{6}] that for each σ = 2, 3, the cubic terms of E_{σ} can be eliminated by a transformation of the form z̃_{σ} = z_{σ} +p_{σ}, where p_{σ} has weight 3 and does not contribute cubic terms to the other equations.

After these steps, the higher degree terms in (67) have a partial normal form:

(68)  E_{σ} = O(4)
e_{τ} = O(4)
e_{n−1} = (∑_{j=3}^{m−1}(e_{n−1}^{1j}z̄_{1}^{2}x_{j} + e_{n−1}^{2j}z_{1}z̄_{1}x_{j} + e_{n−1}^{3j}z̄_{1}x_{2}x_{j}

+(∑_{ℓ=3}^{m−1} e_{n−1}^{4ℓj}z̄_{1}x_{ℓ})x_{j})) + O(4)

e_{n} = (∑_{j=3}^{m−1}(e_{n}^{1j}z̄_{1}^{2}x_{j} + e_{n}^{2j}z_{1}z̄_{1}x_{j} + e_{n}^{3j}z̄_{1}x_{2}x_{j}

+(∑_{ℓ=3}^{m−1} e_{n}^{4ℓj}z̄_{1}x_{ℓ})x_{j})) + O(4),

where in the above double sums, e_{n−1}^{4ℓj} and e_{n}^{4ℓj} are 0 if ℓ < j. A transformation of the form

z̃_{1} = z_{1} + (∑ p_{1}^{2j}z_{2}z_{j}) + (∑ p_{1}^{ℓj}z_{ℓ}z_{j}) ,
z̃_{n−1} = z_{n−1} + (∑ p_{n−1}^{1j}z_{j}z_{n−1}) + (∑ p_{n−1}^{2j}z_{j}z_{n})

can eliminate all the complex cubic coefficients from e_{n−1}.

A transformation of the form

z̃_{1} = z_{1} + ∑ p_{1}^{1j}z_{1}z_{j},
z̃_{n} = z_{n} + (∑ p_{n}^{1j}z_{j}z_{n−1}) + (∑ p_{n}^{2j}z_{j}z_{n})

can eliminate the e_{n}^{1j}, e_{n}^{2j}, e_{n}^{3j} cubic coefficients from e_{n}. The p_{1}^{1j} coefficients re-introduce z̄_{1}^{2}x_{j} terms in e_{n−1}, but they can be eliminated by another z̃_{n−1} = z_{n−1} + ∑ p_{n−1}^{1j}z_{j}z_{n−1} transformation without changing

e_{n}. The partial sum ∑_{ℓ=4}^{m−1} e_{n}^{4ℓj}z̄_{1}x_{ℓ} can be re-grouped exactly as in (66),

so that for each j = 3, …, m − 1, terms of the form e_{n}^{4ℓj}z̄_{1}x_{ℓ}x_{j}, with ℓ = 4, …, m − 1, can be eliminated by a transformation of the form

z̃_{2} = z_{2} + ∑_{ℓ=4}^{m−1} p_{2}^{ℓj}z_{ℓ}z_{j}

z̃_{n} = z_{n} + ∑_{τ=m}^{n−2} p_{n}^{3τj}z_{τ}z_{j},

where the coefficients p_{2}^{ℓj} are real.

Here we are again using the assumption m = \frac{2}{3}(n+1). For \frac{2}{3}(n+1) < m < n, there would be terms z̄_{1}x_{ℓ}x_{j}, with ℓ = 4, …, 2n − 2m + 1, 2n − 2m + 2, …, m − 1, and only enough z_{τ} quantities available to cancel ℓ = 4, …, 2n − 2m + 1.

This leaves the cubic term e_{n}^{433}z̄_{1}x_{3}^{2}. A transformation of the form

z̃ = (a_{1}z_{1}, a_{1}z_{2} + p_{2}^{33}z_{3}^{2}, r_{3}^{3}z_{3}, a_{1}z_{4}, …, a_{1}z_{m−1}, (a_{1})^{2}z_{m}, …, (a_{1})^{2}z_{n})^{T} ,

with a_{1} and r_{3}^{3} nonzero and real, and p_{2}^{33} real, results in the new defining equation

z̃_{n} = z̃_{1} ¯z̃_{1} + ¯ z̃_{1}x̃_{2} + (\frac{a_{1}e_{n}^{433} − p_{2}^{33}}{(r_{3}^{3})^{2}} ) ¯ z̃_{1}x̃_{3}^{2} + O(4),

without introducing any other cubic terms or changing the quadratic part of the other equations. There are no other linear or nonlinear transformations that contribute to this term (the lengthy check is omitted). Using p_{2}^{33} to cancel the real part of the e_{n}^{433} coefficient, and then re-scaling by a_{1}, e_{n}^{433} can be normalized to either i or 0. We regard the 0 case as another degeneracy, and from this point only consider the following cubic normal form:

(69)  y_{σ} = 0 + E_{σ}(z_{1}, z̄_{1}, x) = O(4)
z_{τ} = (x_{2(τ−m+2)} + ix_{2(τ−m+2)+1})z̄_{1} + e_{τ}(z_{1}, z̄_{1}, x)
z_{n−1} = z̄_{1}^{2} + e_{n−1}(z_{1}, z̄_{1}, x),
z_{n} = (z_{1} + x_{2} + ix_{3}^{2})z̄_{1} + e_{n}(z_{1}, z̄_{1}, x),

where E_{2}, …, e_{n} have degree d ≥ 4 and are real analytic.

Proposition 6.2. Given \frac{2}{3}(n+1) = m < n, if M is a real analytic m-submanifold of ℂ^{n} with a CR singularity that can be put into the cubic normal form (69), then there exists a holomorphic coordinate change z̃ = Az⃗+p⃗ as in (9), in a neighborhood of 0 ∈ ℂ^{n}, transforming the defining equations into the following real algebraic normal form:

(70) ỹ_{σ} = 0, σ = 2, …, m − 1
z̃_{τ} = (x̃_{2(τ−m+2)} + ix̃_{2(τ−m+2)+1})¯z̃_{1}, τ = m …, n − 2
z̃_{n−1} = ¯ z̃_{1}^{2}
z̃_{n} = (z̃_{1} + x̃_{2} + ix̃_{3}^{2})¯z̃_{1}.

This Proposition will follow as a corollary of a more general analytic normal form result, Main Theorem 6.5.

Observe that the the involution z̃_{3} = −z_{3}, which reverses the orientation of the tangent plane T0M, leaves the normal form (69) invariant, and is a symmetry of the real variety in Proposition 6.2.

6.2. Unfolding CR singularities of m-submanifolds.

We continue to work with \frac{2}{3}(n + 1) ≤ m < n. Recall that the nondegenerate CR singularities of M are stable in two senses — near such a point, any small real analytic perturbation of M is locally equivalent to a constant normal form, and when m = \frac{2}{3}(n + 1), the intersection index (±1) is also constant under a small smooth perturbation. However, pairs of isolated points with opposite indices may cancel under a larger-scale homotopy. These three phenomena are all reflected in the analysis of the local normal forms for unfoldings — any unfolding of a stable CR singular point will be shown to be u-equivalent to a trivial (or “constant”) unfolding, and a degenerate point will have an unfolding exhibiting a pair creation/annihilation. Unlike the unfoldings of surfaces considered in Section 5, we will find normal forms not only to arbitrarily high degree, but we will find the whole u-equivalence classes for two different normal forms for real analytic submanifolds under the group U_{m,n,k} of local biholomorphic transformations. The formal problem, finding a normal form for higher degree terms, is, like the calculations of Section 5, a matter of comparison of coefficients, to find series expressions for the nonlinear part p⃗ of transformations in U_{m,n,k} (11) in terms of the series expressions for the given defining equations (10). A normalizing transformation may not be unique, so our construction of p⃗ does more than just establish the existence of a formal series, it also makes some choices to avoid series that are divergent or that converge on insufficiently large sets.

Theorem 6.3. Given \frac{2}{3}(n + 1) ≤ m < n, if M is a real analytic msubmanifold of ℂ^{n} with a nondegenerate quadratic normal form as in the hypothesis of Proposition 6.1, and ; M is any real analytic unfolding of M in ℂ^{n+k}, with defining equations in standard position

(71) y_{σ} = E_{σ}(z_{1}, z̄_{1}, x, t) = O(2)
z_{τ} = (x_{2(τ−m+2)} + ix_{2(τ−m+2)+1})z̄_{1}

+e_{τ}^{10α}t_{α}z_{1}+ e_{τ}^{01α}t_{α}z̄_{1}+ e_{τ}^{αβ}t_{α}t_{β} +f_{τ}^{σ_{1}α}x_{σ_{1}}t_{α}+ e_{τ}(z_{1}, z̄_{1}, x, t)

z_{n−1} = z̄_{1}^{2} + e_{n−1}^{10α}t_{α}z_{1} + e_{n−1}^{01α}t_{α}z̄_{1}
+e_{n−1}^{αβ}t_{α}t_{β} + f_{n−1}^{σ_{1}α}x_{σ_{1}}t_{α} + e_{n−1}(z_{1}, z̄_{1}, x, t)
(72) z_{n} = (z_{1} + x_{2} + ix_{3})z̄_{1}

+e_{n}^{10α}t_{α}z_{1}+ e_{n}^{01α}t_{α}z̄_{1}+ e_{n}^{αβ}t_{α}t_{β} +f_{n}^{σ_{1}α}x_{σ_{1}}t_{α}+ e_{n}(z_{1}, z̄_{1}, x, t),

(73) s_{α} = 0, α = 1, …, k

then there exists a holomorphic coordinate change (z̃, w̃) = A(z⃗, w⃗) + p⃗(z⃗, w⃗) in U_{m,n,k} (11), in a neighborhood of (0, 0) ∈ ℂ^{n+k}, transforming the equations (71–72) into the real algebraic normal form (65), which does not depend on t, and preserving the form of (73): s̃_{α} = 0.

Proof. The statement of the Theorem is that any real analytic unfolding ; M of any real analytic, nondegenerate M is u-equivalent (in some small neighborhood of the origin in ℂ^{n+k}) to the trivial unfolding ; M × ℝ^{k}, where ; M is the real algebraic model from the conclusion of Proposition 6.1.

Considering the quadratic part first, as usual any terms in e_{u} not involving z̄_{1} can be eliminated by a transformation of the form z̃_{u} = z_{u} + p_{u}(z_{1}, …, z_{m−1}, w⃗). Similarly, terms z_{1}t_{α} can be eliminated from the real functions E_{σ} in (71) while simultaneously eliminating their complex conjugates. The terms e_{n−1}^{01α}t_{α}z̄_{1} can be eliminated by a linear transformation z̃_{1} = z_{1} + a^{α}w_{α}, as in the γ = ∞ version of Lemma 5.10.

The real and imaginary parts of the coefficients e_{τ}^{01α} and e_{n}^{01α} can be eliminated by real linear transformations of the x variables, z̃_{σ} = z_{σ} + r_{σ}^{α}w_{α}, using real coefficients r_{σ}^{α} from the R_{(m−2)×k} block of the matrix A (12). This is where the second nondegeneracy assumption on the quadratic normal form of M is used.

The result of this transformation is that the quadratic part of the defining equations of the unfolding ; M does not depend on t. In fact, ; M is a submanifold of real dimension m+k in ℂ^{n+k}, satisfying \frac{2}{3}((n+k)+ 1) < (m+k) < n+k, and its quadratic part satisfies the nondegeneracy hypothesis of Proposition 3.3 of [C_{6}], except for a merely notational difference: the real t_{1}, …, t_{k} variables in the above quantities appear in the place of the real x_{m}, …, x_{(m+k)−1} variables from the normal form of [C_{6}], which occur in the higher degree terms but not the quadratic part. The conclusion of that Proposition is that there exists a local holomorphic transformation in B_{m,n+k} of a neighborhood of the origin in ℂ^{n+k},

(74) (z̃, w̃) = (z⃗, w⃗) + (p_{1}(z⃗, w⃗), …, p_{n}(z⃗, w⃗), P_{1}(z⃗, w⃗), …, P_{k}(z⃗, w⃗))^{T} ,

taking ; M to the claimed real algebraic normal form not depending on t and with s̃_{α} = Im(w̃_{α}) = 0. However, that statement is not enough to establish u-equivalence, which requires that there exists such a transformation in the subgroup U_{m,n,k} (11).

An inspection of the Proof of Theorem 5.6 of [C_{6}] will show that the transformation of the x_{σ}, σ = 2(n + k) − 2(m + k) + 2, …, m, …, (m + k) − 1, variables is of the form x_{σ} = Re(z_{σ}), x̃_{σ} = Re(z̃_{σ}), z̃_{σ} = z_{σ} + p_{σ}, such that the coefficients of p_{σ} are linear combinations of the real and imaginary parts of the coefficients from E_{σ}, where y_{σ} = E_{σ} is one of the defining equations of the manifold (in [C_{6}], this is worked out for σ = 6). In the case of ; M, y_{σ}, σ = m, …, (m+k)−1, corresponds to Im(w_{α}) = s_{α}, α = σ − (m − 1) = 1, …, k, and E_{σ} corresponds to the constant function 0 from the RHS of (73). We can conclude that the component w̃_{α} = w_{α} + P_{α}(z⃗, w⃗) of the transformation (74) is simply w̃_{α} = w_{α}. The transformation constructed in Theorem 5.6 and Corollary 5.7 of [C_{6}] conveniently satisfies the reality conditions of (11) as a consequence of the assumption ; M ⊆ ℂ^{n} × ℝ^{k} ⊆ ℂ^{n+k}, and the transformation of Proposition 3.3 of [C_{6}] is the limit of composites of such transformations, so it also is in the subgroup U_{m,n,k}.

In [C_{6}], an analogy was offered between the nondegenerate normal form result (quoted here previously as Proposition 6.1) and a result of Whitney, that the cross-cap parametrization ℝ^{2} → ℝ^{3} : (u, v) ↦ (u, uv, v^{2}) is a normal form for nondegenerate singularities of smooth maps under smooth coordinate changes. The normal form result of the above Theorem 6.3 could similarly be considered as analogous to the cross-cap map’s property that any smooth unfolding is equivalent to a trivial unfolding (in a specific sense; see Part 4 of [Martinet]).

Next, for m = \frac{2}{3}(n + 1), let M have a degenerate CR singularity with cubic normal form (69), and let ; M be a k-parameter unfolding of M in standard position. The defining equations of ; M are the same as (71–72) from Theorem 6.3, except that the term iz̄_{1}x_{3} in (72) is replaced by iz̄_{1}x_{3}^{2}. The elimination of all the quadratic terms involving t proceeds as in the above Proof, with the exception of the e_{n}^{01α}z̄_{1}t_{α} terms. The real part of the complex coefficients e_{n}^{01α} can be eliminated by z̃_{2} = z_{2} + r_{2}^{α}w_{α}, but this leaves the transformed equation for z_{n} in the form

z_{n} = (z_{1} + x_{2} + ix_{3}^{2})z̄_{1} + e_{n}^{01α}t_{α}z̄_{1} + e_{n}(z_{1}, z̄_{1}, x, t),

with purely imaginary e_{n}^{01α}. A real linear transformation of the t variables (the block R_{k×k} of the matrix A from (12)) can normalize the coefficient vector (e_{n}^{011}, …, e_{n}^{01k}) to either (i, 0, …, 0) or (0, …, 0). We regard the zero case as a degenerate unfolding of M, and it is not considered further. A nondegenerate unfolding of M as in (69) has the following normal form:

(75)  y_{σ} = E_{σ}(z_{1}, z̄_{1}, x, t)
z_{τ} = (x_{2(τ−m+2)} + ix_{2(τ−m+2)+1})z̄_{1} + e_{τ}(z_{1}, z̄_{1}, x, t)
z_{n−1} = z̄_{1}^{2} + e_{n−1}(z_{1}, z̄_{1}, x, t)
z_{n} = (z_{1} + x_{2} + it_{1} + ix_{3}^{2})z̄_{1} + e_{n}(z_{1}, z̄_{1}, x, t)
s_{α} = 0

where E_{2}, …, e_{n} have degree 3, and all the cubic terms except iz̄_{1}x_{3}^{2} depend on t.

Remark 6.4. Our first observation on the geometry of ; M in the normal form (75) is that, as a real (m+k)-submanifold of ℂ^{n+k}, it has a nondegenerate CR singularity and is equivalent under a biholomorphic coordinate change in B_{m,n+k} near the origin of ℂ^{n+k} to the real algebraic normal form ; M ×ℝ^{k} from the conclusion of Theorem 6.3. In particular, the complex linear transformation that switches x_{3} and t_{1}, z̃_{3} = w_{1}, w̃_{1} = z_{3}, puts ; M into the same quadratic normal form as (71–73), and then the cubic term i¯z̃_{1}t̃_{1}^{2} can be eliminated as in Proposition 3.3 of [C_{6}]. However, this ; M is not u-equivalent to ; M × ℝ^{k}. Under the action of the subgroup U_{m,n,k} (11), which does not include any linear transformation switching x_{3} and t_{1}, and which preserves the fiber ℂ^{n} × {0}, the slice M = M_{0} retains its degenerate normal form. This is a different phenomenon from Examples 5.23 and 5.24, where surfaces with parabolic points had unfoldings ; M and ; M^{±} that themselves had parabolic points as submanifolds of ℂ^{2+k}.

The following result is the Main Theorem, which states that any real analytic, nondegenerate unfolding (75) of M not only has the algebraizability property under u-equivalence, but, further, is u-equivalent to a unique real algebraic model. The slice M_{0} is simultaneously transformed into an algebraic normal form, so Proposition 6.2 follows as a corollary. The m = 4, k = 1, d = 4 case appeared in the Introduction.

Main Theorem 6.5. Given \frac{2}{3}(n + 1) = m < n, if M is a real analytic m-submanifold of ℂ^{n} with a degenerate CR singularity having cubic normal form (69), and ; M is a nondegenerate real analytic unfolding of M in ℂ^{n+k}, with defining equations in standard position given by (75), then there exists a holomorphic coordinate change (z̃, w̃) = A(z⃗, w⃗) + p⃗(z⃗, w⃗) in U_{m,n,k} (11), in a neighborhood of (0, 0) ∈ ℂ^{n+k}, transforming the equations (75) into the real algebraic normal form:

(76)  ỹ_{σ} = 0
z̃_{τ} = (x̃_{2(τ−m+2)} + ix̃_{2(τ−m+2)+1})¯z̃_{1}
z̃_{n−1} = ¯ z̃_{1}^{2}
z̃_{n} = (z̃_{1} + x̃_{2} + it̃_{1} + ix̃_{3}^{2})¯z̃_{1}
s̃_{α} = 0.

Remark 6.6. The proof of the Main Theorem will be divided into several steps. The first step would be to show there exists a coordinate transformation (z̃, w̃) = A(z⃗, w⃗) + p⃗(z⃗, w⃗) in U_{m,n,k} eliminating all the cubic terms from the normal form (75), except iz̄_{1}x_{3}^{2}, so that the quantities E_{2}, …, E_{m−1} , e_{m}, …, e_{n} are all O(4) in z_{1}, z̄_{1}, x, t. However, rather than going through a lengthy but straightforward calculation to find a suitable invertible matrix A and polynomial p⃗(z⃗, w⃗) achieving such a transformation, we will skip ahead to the next step in an induction, and show that if E_{2}, …, e_{n} are O(d) with d ≥ 4, then there is a holomorphic transformation so that in the new coordinates, the higher degree terms of the defining functions have degree greater than d. The main part of that argument will be the solution of a certain system of linear equations, Theorem 7.6. After the Proof of that Theorem, it will be shown in Remark 7.7 how the calculation of that Proof can be modified in a small way to go back to the start of the induction at d = 3 and to establish this cubic normal form.

Then, showing that there exist a linear part A and a holomorphic function p⃗(z⃗, w⃗), so that (z̃, w̃) = A(z⃗, w⃗) + p⃗(z⃗, w⃗) is in the subgroup U_{m,n,k} and takes the cubic normal form and eliminates all its higher degree terms to transform ; M into the algebraic variety (76), involves finding p⃗ as a solution of a system of nonlinear functional equations, by iterating the solution of the linear system. The convergence argument is given in Section 7 in a series of steps, each stated as a Theorem.

6.3. The geometry of the algebraic unfolding.

Before proceeding to the Proof of Main Theorem 6.5, we conclude this Section by considering the geometry of the real algebraic variety given by (76), denoted ; M, which is a representative of the u-equivalence class for the nondegenerate unfolding ; M of any real analytic M with cubic normal form (69). The example (1) from the Introduction is the

m = 4, n = 5, k = 1 case of ; M.

Recalling the complexification construction from Section 4, corresponding to the smooth real variety ; M ⊆ ℂ^{n+k} is a smooth complex variety ; M_{c} ⊆ ℂ^{2(n+k)}, parametrized by Σ : ℂ^{m+k} → ℂ^{2(n+k)}. Let ℂ^{m+k} have coordinates

(z_{1}, ζ_{1}, ξ_{2}, …, ξ_{m−1}, ω_{1}, …, ω_{k}) = (z_{1}, ζ_{1}, ξ, ω).

The image of ; M_{c} under the projection π : ℂ^{2(n+k)} → ℂ^{n+k} is a singular complex variety, parametrized by π ∘ Σ as in (19):

π ∘ Σ : ℂ^{m+k} → ℂ^{n+k} :
(z_{1}, ζ_{1}, ξ, ω) ↦ (z_{1}, ξ_{2}, …, ξ_{m−1},
…, (ξ_{2(τ−m+1)} + iξ_{2(τ−m+1)+1})ζ_{1}, …,
ζ_{1}^{2}, (z_{1} + ξ_{2} + iω_{1} + iξ_{3}^{2})ζ_{1}, ω_{1}, …, ω_{k})^{T} .

At each point in the domain ℂ^{m+k}, the complex Jacobian of this polynomial map either will have full rank, m + k, or will be singular, with rank m + k − 1. The singular locus in the domain is the affine variety

{ξ_{2(τ−m+1)} + iξ_{2(τ−m+1)+1} = 0, ζ_{1} = 0, z_{1} + ξ_{2} + iω_{1} + iξ_{3}^{2} = 0}.

As in (18), let δ denote the inclusion of the totally real subspace

\{ζ_{1} = z̄_{1}, ξ_{2} = \overline{ξ_{2}} = x_{2}, …, ω_{k} = \overline{ω_{k}} = t_{k}\}

in ℂ^{m+k}, so ; M is the image of π ∘ Σ ∘ δ. The intersection of this real subspace with the singular locus is

{x_{4} = … = x_{m−1} = 0, z_{1} = z̄_{1} = 0, x_{2} = 0, t_{1} + x_{3}^{2} = 0},

so the candidates for CR singular points in ; M are of the form

{(0, 0, x_{3}, 0, …, 0, t_{1}, t_{2}, …, t_{k})^{T} : t_{1} + x_{3}^{2} = 0} ⊆ ; M ⊆ ℂ^{n+k},

and all the other points in ; M are totally real. When k = 1, this is a real parabola in ; M contained in the real x_{3}, t_{1} coordinate plane and tangent to the real x_{3}-axis at the origin; for k > 1, it is a k-dimensional real parabolic cylinder. As observed in Remark 6.4, ; M considered as a real submanifold of ℂ^{n+k} near the origin is locally biholomorphically equivalent to ; M × ℝ^{k}, so it has a totally real k-dimensional locus N_{1} of nondegenerate CR singular points in a neighborhood of (0, 0).

We continue studying the geometry of ; M by considering the slices M_{t} for various t. The slice M_{0} is exactly the normal form variety of Proposition 6.2; as a submanifold of ℂ^{n}, it has a degenerate CR singular point at the origin and is totally real at every other point. For t = (t_{1}, …, t_{n}) with t_{1} > 0, the slice M_{t} is totally real.

For t with t_{1} < 0, the slice M_{t} has two candidates for CR singularities, at (0, 0, x_{3}, 0, …, 0)^{T} ∈ ℂ^{n}, where x_{3} = ±\sqrt{−t_{1}}. Keeping in mind that t is fixed, so t_{1}, …, t_{k} are constants with t_{1} negative, the equations for M_{t} in ℂ^{n} are

y_{σ} = 0
z_{τ} = (x_{2(τ−m+2)} + ix_{2(τ−m+2)+1})z̄_{1}
z_{n−1} = z̄_{1}^{2}
(77)  z_{n} = (z_{1} + x_{2} + it_{1} + ix_{3}^{2})z̄_{1}.

This contains the origin but is not in standard position. Replacing x_{3} with the quantity x_{3}−\sqrt{−t_{1}} is a translation that moves a CR singularity candidate point to the origin, and Equation (77) becomes

z_{n} = (z_{1} + x_{2} − 2i\sqrt{−t_{1}}x_{3} + ix_{3}^{2})z̄_{1},

which is in standard position, in the quadratic normal form (59–61), with ∑ ϵ_{n}^{σ_{1}}x_{σ_{1}} = x_{2} − 2i\sqrt{−t_{1}}x_{3} and e_{n} = ix_{3}^{2}z̄_{1}.

Similarly, the other translation x_{3} ↦ x_{3} + \sqrt{−t_{1}} puts M_{t} into standard position with defining equation

z_{n} = (z_{1} + x_{2} + 2i\sqrt{−t_{1}}x_{3} + ix_{3}^{2})z̄_{1}.

The conclusion is that both candidate points are in fact CR singularities, satisfying both nondegeneracy conditions, with complex tangent lines parallel to the z_{1}-axis, and with opposite indices, ±1, corresponding to the sign of the coefficient of ix_{3}z̄_{1} in the quadratic normal form.

In the k = 1 case, we can consider t_{1} as a time parameter increasing from negative to positive; the unfolding ; M represents a cancellation, where M_{t} has two nondegenerate CR singularities of opposite index that approach each other, meet at the origin at t_{1} = 0 so that M_{0} has a degenerate CR singularity, and then M_{t} is totally real afterward. By Main Theorem 6.5, the same phenomenon is expected to hold, locally in space and for a short interval of time, for any real analytic unfolding ; M (satisfying a nondegeneracy condition as in (75)) of any real analytic M_{0} with a degenerate CR singularity having cubic normal form (69).

7. Rapid convergence Proof of the Main Theorem

7.1. A functional equation.

To show the existence of a normalizing transformation as claimed in Main Theorem 6.5, we will set up a system of nonlinear functional equations, so that any solution p⃗ of the system will define a normalizing transformation (z̃, w̃) = (z⃗, w⃗) + p⃗ in U_{m,n,k} (11). In addition to finding a formal power series solution, we will also have to show that the solution is convergent in some neighborhood of the origin. The method of proof is the rapid convergence technique, as used in [Moser], [C_{4}], and [C_{6}]. Rather than trying to solve the system of equations directly, we first find an approximate solution by solving a related system of linear equations. Iteration of this process gives a sequence of approximations that approach an exact solution. We are careful to construct the approximations so that their domains shrink slowly enough so their diameters are bounded below by a positive constant.

We start by considering the effect of a coordinate change (11) on the quadratic and cubic parts of the defining equations in normal form (75). The transformation (z̃, w̃) = (z⃗, w⃗) + p⃗ is formally invertible near (0, 0), and it may be useful to think of (z̃, w̃) = (z⃗, w⃗) + p⃗ as having identity linear part, although there may be some linear terms in p⃗, which could be included as entries in the above-diagonal blocks in the matrix A (12). In terms of z̃, w̃, z⃗, w⃗, consider the system of equations

(78) 0 = Im(z̃_{σ}) = Im(z_{σ} + p_{σ}(z⃗, w⃗))
0 = z̃_{τ} − (¯z̃_{1}x̃_{2(τ−m+2)} + i¯z̃_{1}x̃_{2(τ−m+2)+1})
= z_{τ} + p_{τ}(z⃗, w⃗)
−\overline{(z_{1} + p_{1}(z⃗, w⃗))} · Re(z_{2(τ−m+2)} + p_{2(τ−m+2)}(z⃗, w⃗))

−i\overline{(z_{1} + p_{1}(z⃗, w⃗))} · Re(z_{2(τ−m+2)+1} + p_{2(τ−m+2)+1}(z⃗, w⃗))

0 = z̃_{n−1} − ¯ z̃_{1}^{2}
= z_{n−1} + p_{n−1}(z⃗, w⃗) − \overline{(z_{1} + p_{1}(z⃗, w⃗))}^{2}
0 = z̃_{n} − (z̃_{1} + x̃_{2} + it̃_{1} + ix̃_{3}^{2})¯z̃_{1}
= z_{n} + p_{n}(z⃗, w⃗)
−\overline{(z_{1} + p_{1}(z⃗, w⃗))} · (z_{1} + p_{1}(z⃗, w⃗))
−\overline{(z_{1} + p_{1}(z⃗, w⃗))} · Re(z_{2} + p_{2}(z⃗, w⃗))
−i\overline{(z_{1} + p_{1}(z⃗, w⃗))} · Re(w_{1} + P_{1}(w⃗))
−i\overline{(z_{1} + p_{1}(z⃗, w⃗))} · (Re(z_{3} + p_{3}(z⃗, w⃗)))^{2}
0 = s̃_{α} = Im(w̃_{α}) = Im(w_{α} + P_{α}(w⃗)).

In order to get (76) to be the defining equations for ; M in the (z̃, w̃) coordinates, the above equalities must hold for points (z⃗, w⃗) on ; M and near (0, 0). So, we can replace the z⃗ = (z_{1}, …, z_{n})^{T} and w⃗ = (w_{1}, …, w_{k})^{T} expressions in (78) by the defining functions (75):

(79)  z⃗ = (z_{1}, x_{2} + iE_{2}(z_{1}, z̄_{1}, x, t), …, h_{n}(z_{1}, z̄_{1}, x, t))^{T} ,
w⃗ = t = (t_{1}, …, t_{k}),

to get a system of equations where the RHS functions depend only on z_{1}, z̄_{1}, x, t:

(80) 0 = Im(x_{σ} + iE_{σ} + p_{σ}(z⃗, t)) = E_{σ}(z_{1}, z̄_{1}, x, t) + Im(p_{σ}(z⃗, t))
0 = e_{τ}(z_{1}, z̄_{1}, x, t) + p_{τ}(z⃗, t)
−\overline{p_{1}(z⃗, t)} · (x_{2(τ−m+2)} + ix_{2(τ−m+2)+1})
−z̄_{1} · (Re(p_{2(τ−m+2)}(z⃗, t)) + iRe(p_{2(τ−m+2)+1}(z⃗, t)))
−\overline{p_{1}(z⃗, t)} · (Re(p_{2(τ−m+2)}(z⃗, t)) + iRe(p_{2(τ−m+2)+1}(z⃗, t)))
0 = e_{n−1}(z_{1}, z̄_{1}, x, t) + p_{n−1}(z⃗, t) − 2z̄_{1}\overline{p_{1}(z⃗, t)} − \overline{p_{1}(z⃗, t)}^{2}
0 = e_{n}(z_{1}, z̄_{1}, x, t) + p_{n}(z⃗, t)
−z̄_{1} · (p_{1}(z⃗, t) + Re(p_{2}(z⃗, t)) + iP_{1}(t) + 2ix_{3}Re(p_{3}(z⃗, t)))

−iz̄_{1} · (Re(p_{3}(z⃗, t)))^{2} (81)

−\overline{(p_{1}(z⃗, t))} · (z_{1} + x_{2} + it_{1} + ix_{3}^{2})
−\overline{(p_{1}(z⃗, t))} · (p_{1}(z⃗, t) + Re(p_{2}(z⃗, t)) + iP_{1}(t))
−\overline{(p_{1}(z⃗, t))} · (2ix_{3}Re(p_{3}(z⃗, t)) + i(Re(p_{3}(z⃗, t))^{2}))
(82) 0 = Im(t_{α} + P_{α}(t)).

The components of e⃗ = (E_{2}, …, E_{m−1}, e_{m}, …, e_{n}) appear in (80–82) in two ways — as terms in the equations, and also in the z⃗ input (79) for each p_{j}(z⃗, t) in (80–82), j = 1, …, n. So, given e⃗, if we happen to have an exact solution p⃗ of the above system of functional equations, the conclusion of Main Theorem 6.5 holds and we are done. However, (80– 82) is a nonlinear system in the unknown quantity p⃗, where in addition to the composition with the given defining functions (79), there are products of the components P_{1}, p_{j}, and \overline{p_{j}}.

If k > 1, there are unknown quantities P_{2}(t), …, P_{k}(t), that appear only in line (82). The quantity Im(t_{α} + P_{α}(t)) is identically zero for any P_{α} satisfying the reality condition of the subgroup U_{m,n,k}, so we can choose P_{2} = … = P_{k} = 0 as the solution, leaving only the real quantity P_{1}(t) as an unknown to be determined in terms of e⃗.

Suppose e⃗ has degree d ≥ 3, then by inspection of the system (79– 82), the components of a solution p⃗ have lower bounds on the weight: p_{m}, …, p_{n} have weight ≥ d, P_{1}, p_{1}, p_{2}, p_{4}, …, p_{m−1} have weight ≥ d−1, and p_{3} has weight ≥ d − 2. For d = 3, p_{3} may include some weight 1 linear terms, of the form

(83)  z̃_{3} = z_{3} + (∑ r_{3}^{α}t_{α}) + (weight ≥ 2),

where r_{3}^{α} entries from the block R_{(m−2)×k} from matrix A (12) will be needed (in Remark 7.7) to cancel the imaginary parts of cubic terms of the form e_{n}^{α}z̄_{1}x_{3}t_{α}.

As a first step in solving for p⃗ = (p_{1}, …, p_{n}, P_{1}, 0, …, 0)^{T} in terms of e⃗, consider the following system of simpler equations:

(84) 0 = E_{σ}(z_{1}, z̄_{1}, x, t) + Im(p_{σ}(z⃗, t))
(85) 0 = e_{τ}(z_{1}, z̄_{1}, x, t) + p_{τ}(z⃗, t)
−\overline{p_{1}(z⃗, t)} · (x_{2(τ−m+2)} + ix_{2(τ−m+2)+1})
−z̄_{1} · (Re(p_{2(τ−m+2)}(z⃗, t)) + iRe(p_{2(τ−m+2)+1}(z⃗, t)))
(86) 0 = e_{n−1}(z_{1}, z̄_{1}, x, t) + p_{n−1}(z⃗, t) − 2z̄_{1}\overline{p_{1}(z⃗, t)}
(87) 0 = e_{n}(z_{1}, z̄_{1}, x, t) + p_{n}(z⃗, t)

−z̄_{1} · (p_{1}(z⃗, t) + Re(p_{2}(z⃗, t)) + iP_{1}(t) + 2ix_{3}Re(p_{3}(z⃗, t)))

−\overline{(p_{1}(z⃗, t))} · (z_{1} + x_{2} + it_{1} + ix_{3}^{2}),

where the z⃗ input for each p_{j} is:

(88) z⃗ = (z_{1}, x_{2}, …, x_{m−1}, (x_{4}+ix_{5})z̄_{1}, …, z̄_{1}^{2}, (z_{1}+x_{2}+it_{1}+ix_{3}^{2})z̄_{1})^{T} .

This simplifies p_{j}(z⃗, t) by considering only the lower degree terms of the input (79) that appear in the algebraic normal form (76). Also, the products of P_{1}, p_{j}, \overline{p_{j}} are dropped, so that these are (real) linear equations.

To see how the new equations are related to the original system, suppose e⃗ has degree d ≥ 3, and that p⃗ is a solution of (84–88) so that p_{3} has weight ≥ d − 2, p_{1}, p_{2}, p_{4}, …, p_{m−1} have weight ≥ d − 1, and p_{m}, …, p_{n} have weight ≥ d. Evaluating the RHS of (80–82) with this solution for p⃗ results in expressions of degree ≥ 2d − 2, except for the −iz̄_{1} · (Re(p_{3}(z⃗, t)))^{2} quantity (81), which has degree 2d − 3 if p_{3} has weight d − 2. Converting these expressions in z_{1}, z̄_{1}, x, t to z̃_{1}, ¯ z̃_{1}, x̃, t̃, and equating them to the z̃ expressions in (78) gives the higher degree terms of the new defining equations for ; M in the (z̃, w̃) coordinate system. (It will be shown later (Theorem 7.13) that in fact for (z⃗, w⃗) ∈ ; M close enough to (0, 0), z_{1}, z̄_{1}, x, t are real analytic functions of z̃_{1}, ¯ z̃_{1}, x̃, t̃.) If d ≥ 4, then 2d − 3 > d, so the expressions E_{2}, …, e_{n} increase in degree, and while a solution p⃗ of the linearized equations is just an approximation to the solution of the original system, using such a p⃗ to define a coordinate transformation does have the effect of nearly doubling the order of vanishing of the e⃗ quantity.

7.2. A solution of a linearized equation.

The goal of this Subsection is to construct a solution p⃗ of the system of linear equations (84–88), given the higher degree terms of the defining equations, e⃗. The solution p⃗ constructed here will be an (n+k)-tuple of series in z⃗ = (z_{1}, …, z_{n})^{T} and w⃗ = (w_{1}, …, w_{k})^{T} with the following properties: the transformation (z̃, w̃) = (z⃗, w⃗) + p⃗(z⃗, w⃗) is in the subgroup U_{m,n,k} (11), the size of the domain of convergence of p⃗ in ℂ^{n+k} is comparable in a certain sense to the size of the domain of e⃗, and also a suitable norm of p⃗ is bounded in terms of a suitable norm of e⃗.

Notation 7.1. For r = (r_{1}, …, r_{N}) ∈ ℝ^{N}, with all r_{j} > 0, define a polydisc in ℂ^{N} by

𝔻_{r} = {(Z_{1}, …, Z_{N}) : |Z_{j}| < r_{j}}.

As special cases, let

D_{r} = 𝔻_{(r,r,...,r)} ⊆ ℂ^{m+k}

and

Δ_{r} = 𝔻_{(r,...,r,2r^{2},...,2r^{2},r^{2},3r^{2}+r^{3},r,...,r)} ⊆ ℂ^{n+k},

where there are m − 1 radius lengths r, followed by n − m − 1 radius lengths 2r^{2}, in the z_{m}, …, z_{n−2} coordinate directions, and k radius lengths r in the w⃗ directions.

The initial assumption on the defining equations (75) is that

e⃗(z_{1}, z̄_{1}, x, t) = (E_{2}, …, E_{m−1}, e_{m}, …, e_{n})

is real analytic, so there is some r > 0 so that each component of e⃗ is the restriction to {ζ = z̄_{1}, x = x̄, t = t̄} of a multivariable power series in (z_{1}, ζ, x, t) centered at the origin of ℂ^{m+k}, and with complex coefficients, which converges on a complex polydisc D_{r} ⊆ ℂ^{m+k} (or, equivalently, a complex analytic function on D_{r}).

Notation 7.2. For a complex valued function e(z_{1}, ζ, x, t) of m+k complex variables, which is defined on some set containing the polydisc D_{r}, define the norm

|e|_{r} = sup_{(z_{1},ζ,x,t)∈D_{r}} |e(z_{1}, ζ, x, t)|.

For an (n − 1)-tuple e⃗ = (E_{2}, …, e_{n}), define

|e⃗|_{r} = |E_{2}|_{r} + ⋯ + |e_{n}|_{r}.

For a complex valued function p(z_{1}, …, z_{n}, w_{1}, …, w_{k}) of n+k complex variables, which is defined on some set containing the polydisc Δ_{r}, define the norm

∥p∥_{r} = sup_{(z⃗,w⃗)∈Δ_{r}} |p(z⃗, w⃗)|.

With this notation, we can further assume r > 0 is small enough so that |e⃗(z_{1}, ζ, x, t)|_{r} is finite. Given e⃗ with degree ≥ 4, the eventual goal is to find some r̃, 0 < r̃ ≤ r, and a holomorphic map p⃗ : Δr̃ → ℂ^{n+k}, so that the transformation (z̃, w̃) = (z⃗, w⃗) + p⃗(z⃗, w⃗) is a biholomorphism with domain Δr̃ taking ; M to ; M with the algebraic normal form (76). That is, if (z⃗, w⃗) ∈ ; M ∩Δr̃, then (z̃, w̃) satisfies (76). However, for now we are only looking for p⃗ that is a solution of (84–88).

Some steps of the Proof of Theorem 7.6 will decompose series into subseries and their complex conjugates, where these preliminary Lemmas on the |e|_{r} norm will be useful.

Lemma 7.3. Given 0 < R < r and complex coefficients f^{abIK}, g^{abIK}, if

|∑ f^{abIK}z_{1}^{a}ζ^{b}x^{I}t^{K}|_{r} ≤ K

and for complex x and t with |x_{σ}| < r, |t_{α}| < r, a, b = 0, 1, 2, 3, …,

|∑_{I,K} g^{abIK}x^{I}t^{K}| ≤ |∑_{I,K} f^{abIK}x^{I}t^{K}| ,

then

|∑ g^{abIK}z_{1}^{a}ζ^{b}x^{I}t^{K}|_{R} ≤ \frac{Kr^{2}}{(r − R)^{2}} .

This is Lemma 5.3 of [C_{6}], the Proof of which uses Cauchy’s Estimate and the geometric series formula.

In the applications of the Lemma, for each pair (a, b), the coefficients g^{abIK} will either be zero for all I, K, or equal to f^{abIK} for all I, K, so the estimate in the hypothesis is satisfied.

Notation 7.4. On the complex vector space of formal power series, define the following real structure operator:

e = ∑ e^{abIK}z_{1}^{a}ζ^{b}x^{I}t^{K} ↦ e′ = ∑ \overline{e^{abIK}}ζ^{a}z_{1}^{b}x^{I}t^{K}.

The special case h ↦ ℏ of this operation appeared in Section 4.

Lemma 7.5. Given r > 0, the restriction of the above map to the subspace {e : |e|_{r} < ∞} is an isometry.

This is Lemma 5.5 of [C_{6}].

Of course, this map is a representation of complex conjugation: given a series e(z_{1}, z̄_{1}, x, t) for real x and t, which complexifies to e = e(z_{1}, ζ, x, t) for (z_{1}, ζ, x, t) ∈ D_{r} for the purposes of finding its norm as in Notation 7.2, expanding \overline{e(z_{1}, z̄_{1}, x, t)} as a series in (z_{1}, z̄_{1}, x, t) and then complexifying gives e′ = e′(z_{1}, ζ, x, t).

Theorem 7.6. For each m = 4, 6, 8, …, and n satisfying \frac{2}{3}(n+1) = m < n, there are nonzero polynomials C_{m}^{1}(., ., ., .) and C_{m}^{2}(., ., ., .) with real, nonnegative coefficients such that for any 0 < R < r, and any e⃗(z_{1}, ζ, x, t) convergent on D_{r} with |e⃗|_{r} < ∞ and degree d ≥ 4, there exists

p⃗(z⃗, w⃗) = (p_{1}, …, p_{n}, P_{1}, …, P_{k})^{T}

which is convergent on Δ_{r}, and solves the linear system of equations (84–88), and satisfies: P_{2} = … = P_{k} = 0, P_{1}(t) = \overline{P_{1}(t)} for real t, and such that

max{∥p_{1}∥_{R}, …, ∥p_{n}∥_{R}, ∥P_{1}∥_{R}}
≤ (C_{m}^{1}(r, R, \frac{1}{r}, \frac{1}{R}) + \frac{C_{m}^{2}(r, R, \frac{1}{r}, \frac{1}{R})}{(r − R)^{2}}) |e⃗|_{r}.

Proof. First, notice that if p⃗(z_{1}, …, z_{n}, w_{1}, …, w_{k}) is a formal series solution of (84–88), it does not follow that p⃗ is convergent at any point (other than the origin). For example, with any component p_{j}, the series expressions p_{j}(z⃗, w⃗) and

(89)  p_{j}(z⃗, w⃗) + ((z_{1} + z_{2} + iw_{1} + iz_{3}^{2})^{2}z_{n−1} − z_{n}^{2}) · Q(z⃗, w⃗)

are formally the same when restricted to z⃗ as in (88) and w⃗ = t, for any (possibly divergent) series Q. So, if one formal solution p⃗ exists, then there exist infinitely many divergent solutions. There may also exist formal series solutions that are convergent only on some neighborhood of the origin much smaller than that claimed in the Theorem.

Continuing with the abbreviations x = x_{2}, …, x_{m−1}, t = t_{1}, …, t_{k}, and also using z = z_{2}, …, z_{m−1}, and multi-index notation z^{I} and w^{K} = w_{1}^{k_{1}} ⋯ w_{k}^{k_{k}}, the following choice of normalization will simplify the construction of the solution p⃗ satisfying the claimed convergence and bounds:

p_{1}(z⃗, w⃗) = p_{1}(z_{1}, z, z_{n−1}, w⃗)
p_{j}(z⃗, w⃗) = p_{j}^{E}(z_{1}, z, z_{n−1}, w⃗) + z_{n}p_{j}^{O}(z_{1}, z, z_{n−1}, w⃗), j = 2, …, n − 1
p_{n}(z⃗, w⃗) = p_{n}^{E}(z_{1}, z, z_{n−1}, w⃗) + z_{n}p_{n}^{O}(z_{1}, z, z_{n−1}, w⃗)
+p_{n}^{L}(z, z_{m}, …, z_{n−2}, w⃗).

Note that p_{n}^{L} is the only term in the above expressions that depends on z_{m}, …, z_{n−2}, and we will further assume that the quantity p_{n}^{L} is linear in z_{m}, …, z_{n−2}:

p_{n}^{L}(z, z_{m} …, z_{n−2}, w⃗) = ∑_{τ=m}^{n−2} p_{n,τ}^{L}(z, w⃗) · z_{τ}.

The first component p_{1} does not depend on z_{n}, and the remaining components, p_{j}, have some terms not depending on z_{n}, labeled p_{j}^{E}, and other terms which have exactly one linear factor of z_{n}. The p_{j}^{E} and p_{j}^{O} terminology corresponds to even and odd powers of z̄_{1} which appear after the substitution of (88) into p⃗. The choice that p⃗ has at most linear terms in z_{τ} = (x_{2(τ−m+2)}+ix_{2(τ−m+2)+1})z̄_{1} and z_{n} = (z_{1}+x_{2}+it_{1}+ix_{3}^{2})z̄_{1} is made to avoid high powers of the non-monomial quantities, since any multinomial coefficients in the series expansion of p⃗(z⃗, w⃗) could be large enough to affect the size of the domain of convergence.

The exact formulas for the claimed polynomials C_{m}^{1} and C_{m}^{2} are not important, it will be enough that they depend only on m and not on d or e⃗, and that (r − R)^{−2} is the only (nonzero) power of r − R. These properties will follow from the choices made in the construction of p⃗, and from estimates calculated in the course of this Proof.

We begin with (86), the e_{n−1} equation of system (84–88), and apply the normalization condition on p⃗ to get the linear equation:

(90)  0 = e_{n−1}(z_{1}, z̄_{1}, x, t) + p_{n−1}^{E}(z_{1}, x, z̄_{1}^{2}, t)
+(z_{1} + x_{2} + it_{1} + ix_{3}^{2})z̄_{1}p_{n−1}^{O}(z_{1}, x, z̄_{1}^{2}, t)
−2z̄_{1}p_{1}(z_{1}, x, z̄_{1}^{2}, t).

If the series expansion of e_{n−1} had only even powers of z̄_{1}, then it would be a simple matter to compare the coefficients of e_{n−1}(z_{1}, z̄_{1}, x, t) and p_{n−1}^{E}(z_{1}, x, z̄_{1}^{2}, t), and get a solution of the equation with p_{n−1}^{O} = p_{1} = 0. The odd powers of z̄_{1} in e_{n−1} make the p_{n−1}^{O} and p_{1} quantities necessary to solve the equation.

The consideration of the terms of the components of the given quantity e⃗ which are even or odd in z̄_{1} follows the general plan for comparison of coefficients from [Beloshapka], [C_{1}], [C_{4}] in other normal form problems for CR singularities, and from analogous calculations in [Whitney_{1}] for singularities of differentiable maps. We will also use some subseries decompositions, rearrangements, and add-and-subtract tricks, as in [C_{5}], [C_{6}], to get the given terms of e⃗ to correspond to the available terms of p⃗ in (90). In fact, e_{n−1} will be decomposed into more subseries than necessary just to solve Equation (90); the plan is to organize the solution p_{1} into pieces that will be convenient for the later solution of Equation (87).

First, decompose e_{n−1} into even and odd parts e_{n−1,A}, e_{n−1,B}, e_{n−1,C}:

(91) e_{n−1} = ∑ e_{n−1}^{abIK}z_{1}^{a}z̄_{1}^{b}x^{I}t^{K} = e_{n−1,A} + e_{n−1,B} + e_{n−1,C},
e_{n−1,A} = ∑_{b even} e_{n−1}^{abIK}z_{1}^{a}z̄_{1}^{b}x^{I}t^{K}
e_{n−1,B} = ∑_{a even, b odd} e_{n−1}^{abIK}z_{1}^{a}z̄_{1}^{b}x^{I}t^{K}
e_{n−1,C} = ∑_{a, b odd} e_{n−1}^{abIK}z_{1}^{a}z̄_{1}^{b}x^{I}t^{K}.

Since these subseries can be found by averaging, there are the estimates:

|e_{n−1,A}|_{r}

= |\frac{1}{2}(e_{n−1}(z_{1}, ζ, x) + e_{n−1}(z_{1}, −ζ, x))|_{r} ≤ |e_{n−1}|_{r},

|e_{n−1,B}|_{r}

= |\frac{e_{n−1}(z_{1}, ζ, x) − e_{n−1}(z_{1}, −ζ, x) + e_{n−1}(−z_{1}, ζ, x) − e_{n−1}(−z_{1}, −ζ, x)}{4}|_{r}
≤ |e_{n−1}|_{r},

|e_{n−1,C}|_{r}

= |\frac{e_{n−1}(z_{1}, ζ, x) − e_{n−1}(z_{1}, −ζ, x) − e_{n−1}(−z_{1}, ζ, x) + e_{n−1}(−z_{1}, −ζ, x)}{4}|_{r}
≤ |e_{n−1}|_{r}.

Since the x_{2} and x_{3} variables have a distinguished role in the normal form, we will index them separately and use the multi-index J to denote x^{J} = x_{4}^{j_{4}} ⋯ x_{m−1}^{j_{m−1}}. In the (m, n) = (4, 5) case, there are no such variables, but to avoid changing notation for this case, we can just assume J = 0 and x^{J} = 1 in the calculations that follow.

The e_{n−1,B} subseries is further decomposed:

(92) e_{n−1,B} = e_{n−1,D} + e_{n−1,E} + e_{n−1,F} + e_{n−1,G} + e_{n−1,H} + e_{n−1,I},
e_{n−1,D} = ∑_{a even, b odd, d > 0} e_{n−1}^{abcdJK}z_{1}^{a}z̄_{1}^{b}x_{2}^{c}x_{3}^{d}x^{J}t^{K}
e_{n−1,E} = ∑_{a even, b odd, b > 1} e_{n−1}^{abc0JK}z_{1}^{a}z̄_{1}^{b}x_{2}^{c}x^{J}t^{K}
e_{n−1,F} = ∑_{a even, a > 0} e_{n−1}^{a1c0JK}z_{1}^{a}z̄_{1}x_{2}^{c}x^{J}t^{K}
e_{n−1,G} = ∑_{c > 0} e_{n−1}^{01c0JK}z̄_{1}x_{2}^{c}x^{J}t^{K}
e_{n−1,H} = ∑_{J ≠ 0} e_{n−1}^{0100JK}z̄_{1}x^{J}t^{K}
e_{n−1,I} = ∑ e_{n−1}^{01000K}z̄_{1}t^{K}.

(The exponent and summation index ^{d} is not to be confused with the symbol d used to denote the degree of e⃗.)

Each of these subseries can be estimated in terms of |e_{n−1,B}|_{r}:

|e_{n−1,D}|_{r} = |e_{n−1,B}(z_{1}, ζ, x, t) − e_{n−1,B}(z_{1}, ζ, x_{2}, 0, x_{4}, …, x_{m−1}, t)|_{r}
≤ 2|e_{n−1,B}|_{r}.

Let f_{n−1,A} denote the formal series

f_{n−1,A}(z_{1}, ζ, x, t) = \frac{e_{n−1,B}(z_{1}, ζ, x_{2}, 0, x_{4}, …, x_{m−1}, t)}{ζ},

which extends to a holomorphic function f_{n−1,A} on D_{r} since e_{n−1,B} is odd in ζ. Using the notation D_{r}^{∗} = D_{r} \ {ζ = 0}, the maximum principle, and the Schwarz Lemma ([A]),

|f_{n−1,A}|_{r} = sup_{(z,ζ,x,t)∈D_{r}^{∗}} |\frac{e_{n−1,B}(z_{1}, ζ, x_{2}, 0, x_{4}, …, x_{m−1}, t)}{ζ}|

≤ sup_{(z,ζ,x,t)∈D_{r}^{∗}} \frac{\frac{|ζ|}{r} sup_{|ζ|<r} |e_{n−1,B}(z_{1}, ζ, x_{2}, 0, x_{4}, …, x_{m−1}, t)|}{|ζ|}
≤ \frac{1}{r}|e_{n−1,B}|_{r}.

From

e_{n−1,E}(z_{1}, ζ, x, t) = ζ · (f_{n−1,A}(z_{1}, ζ, x, t) − f_{n−1,A}(z_{1}, 0, x, t)),

7. RAPID CONVERGENCE PROOF OF THE MAIN THEOREM 71

we get the estimate |e_{n−1,E}|_{r} ≤ 2|e_{n−1,B}|_{r}. Similarly, from

e_{n−1,F} (z_{1}, ζ, x, t) = ζ · (f_{n−1,A}(z_{1}, 0, x, t) − f_{n−1,A}(0, 0, x, t)), we get the estimate |e_{n−1,F} |_{r} ≤ 2|e_{n−1,B}|_{r}, from

e_{n−1,G}(z_{1}, ζ, x, t)

= ζ · (f_{n−1,A}(0, 0, x_{2}, …, x_{m−1}, t) − f_{n−1,A}(0, 0, 0, 0, x_{4}, …, x_{m−1}, t)),

we get the estimate |e_{n−1,G}|_{r} ≤ 2|e_{n−1,B}|_{r}, from

e_{n−1,H}(z_{1}, ζ, x, t)

= ζ · (f_{n−1,A}(0, 0, 0, 0, x_{4}, …, x_{m−1}, t) − f_{n−1,A}(0, 0, 0, 0, 0, …, 0, t)),

we get the estimate |e_{n−1,H}|_{r} ≤ 2|e_{n−1,B}|_{r}, and from

e_{n−1,I}(z_{1}, ζ, x, t) = ζf_{n−1,A}(0, 0, 0, 0, 0, …, 0, t),

we get the estimate |e_{n−1,I}|_{r} ≤ |e_{n−1,B}|_{r}.

The e_{n−1,C} subseries also decomposes into parts:

(93)  e_{n−1,C} = e_{n−1,J} + e_{n−1,K},
e_{n−1,J} = ∑_{a, b odd, d > 0} e_{n−1}^{abcdJK}z_{1}^{a}z̄_{1}^{b}x_{2}^{c}x_{3}^{d}x^{J}t^{K}
e_{n−1,K} = ∑_{a, b odd} e_{n−1}^{abc0JK}z_{1}^{a}z̄_{1}^{b}x_{2}^{c}x^{J}t^{K},

with each subseries estimated in terms of |e_{n−1,C}|_{r}:

|e_{n−1,J}|_{r} = |e_{n−1,C}(z_{1}, ζ, x, t) − e_{n−1,C}(z_{1}, ζ, x_{2}, 0, x_{4}, …, x_{m−1}, t)|_{r}
≤ 2|e_{n−1,C}|_{r},
|e_{n−1,K}|_{r} = |e_{n−1,C}(z_{1}, ζ, x_{2}, 0, x_{4}, …, x_{m−1}, t)|_{r}
≤ |e_{n−1,C}|_{r}.

72 UNFOLDING CR SINGULARITIES

These two series are further rearranged:

e_{n−1,J} = e_{n−1,L} − e_{n−1,M},
e_{n−1,L} = ∑_{a, b odd, d > 0}e_{n−1}^{abcdJK}z_{1}^{a−1}z̄_{1}^{b−1}x_{2}^{c}x_{3}^{d}x^{J}t^{K} · (z_{1} + x_{2} + it_{1} + ix_{3}^{2})z̄_{1}
e_{n−1,M} = ∑_{a, b odd, d > 0} e_{n−1}^{abcdJK}z_{1}^{a−1}z̄_{1}^{b}x_{2}^{c}x_{3}^{d}x^{J}t^{K} · (x_{2} + it_{1} + ix_{3}^{2}),
e_{n−1,K} = e_{n−1,N} − e_{n−1,O} − e_{n−1,P} − e_{n−1,Q}
−e_{n−1,R} − e_{n−1,S} − e_{n−1,T} − e_{n−1,U},
e_{n−1,N} = ∑_{a, b odd} e_{n−1}^{abc0JK}z_{1}^{a−1}z̄_{1}^{b−1}x_{2}^{c}x^{J}t^{K} · (z_{1} + x_{2} + it_{1} + ix_{3}^{2})z̄_{1}
e_{n−1,O} = ∑_{a, b odd} e_{n−1}^{abc0JK}z_{1}^{a−1}z̄_{1}^{b}x_{2}^{c}x^{J}t^{K} · (ix_{3}^{2})
e_{n−1,P} = ∑_{a, b odd, b > 1} e_{n−1}^{abc0JK}z_{1}^{a−1}z̄_{1}^{b}x_{2}^{c}x^{J}t^{K} · (x_{2} + it_{1})
e_{n−1,Q} = ∑_{a odd, a > 1} e_{n−1}^{a1c0JK}z_{1}^{a−1}z̄_{1}x_{2}^{c}x^{J}t^{K} · (x_{2} + it_{1})
e_{n−1,R} = ∑ e_{n−1}^{11c0JK}z̄_{1}x_{2}^{c}x^{J}t^{K} · (x_{2})
e_{n−1,S} = ∑_{c > 0} e_{n−1}^{11c0JK}z̄_{1}x_{2}^{c}x^{J}t^{K} · (it_{1})
e_{n−1,T} = ∑_{J ≠ 0} e_{n−1}^{1100JK}z̄_{1}x^{J}t^{K} · (it_{1})
e_{n−1,U} = ∑ e_{n−1}^{11000K}z̄_{1}t^{K} · (it_{1}).

The Schwarz Lemma gives an estimate for e_{n−1,M}, and similarly for e_{n−1,O}, where this time D_{r}^{∗} denotes D_{r} \ {z_{1} = 0}:

e_{n−1,M} = \frac{x_{2} + it_{1} + ix_{3}^{2}}{z_{1}} e_{n−1,J}(z_{1}, z̄_{1}, x, t)
⟹ |e_{n−1,M}|_{r} = sup_{D_{r}^{∗}} |(x_{2} + it_{1} + ix_{3}^{2})\frac{e_{n−1,J}(z_{1}, ζ, x, t)}{z_{1}} |
≤ (2r + r^{2}) sup_{D_{r}^{∗}} \frac{\frac{|z_{1}|}{r} sup_{|z_{1}|<r} |e_{n−1,J}(z_{1}, ζ, x, t)|}{|z_{1}|}
≤ (2 + r)|e_{n−1,J}|_{r}.

e_{n−1,O} = \frac{ix_{3}^{2}}{z_{1}} e_{n−1,K}(z_{1}, z̄_{1}, x, t)
⟹ |e_{n−1,O}|_{r} = sup_{D_{r}^{∗}} |(ix_{3}^{2})\frac{e_{n−1,K}(z_{1}, ζ, x, t)}{z_{1}} |

≤ r^{2} sup_{D_{r}^{∗}} \frac{\frac{|z_{1}|}{r} sup_{|z_{1}|<r} |e_{n−1,K}(z_{1}, ζ, x, t)|}{|z_{1}|} ≤ r|e_{n−1,K}|_{r}.

For estimates on the remaining terms, we introduce the expression

f_{n−1,B} = \frac{e_{n−1,K}(z_{1}, ζ, x, t)}{z_{1}ζ},

which can be estimated using two applications of the Schwarz Lemma:

(94) |f_{n−1,B}|_{r} = sup_{(z_{1},ζ,x,t)∈D_{r}^{∗∗}} |\frac{e_{n−1,K}(z_{1}, ζ, x, t)}{z_{1}ζ}|

≤ sup_{(z_{1},ζ,x,t)∈D_{r}^{∗∗}} \frac{\frac{|z_{1}|}{r} sup_{|z_{1}|<r} |e_{n−1,K}(z_{1}, ζ, x, t)|}{|z_{1}||ζ|}

≤ \frac{1}{r} sup_{(z_{1},ζ,x,t)∈D_{r}^{∗∗}} \frac{sup_{|z_{1}|<r} |\frac{|ζ|}{r} sup_{|ζ|<r} |e_{n−1,K}(z_{1}, ζ, x, t)||}{|ζ|}
≤ \frac{1}{r^{2}} |e_{n−1,K}|_{r}.

In some of the above steps, we restricted to the open subset D_{r}^{∗∗} = D_{r} \({z_{1} = 0}∪{ζ = 0}), again to avoid division by 0 but not affecting the sup by the maximum principle.

e_{n−1,P} = (x_{2} + it_{1})ζ · (f_{n−1,B}(z_{1}, ζ, x, t) − f_{n−1,B}(z_{1}, 0, x, t)) ⟹
|e_{n−1,P} |_{r} ≤ 2r^{2}(|f_{n−1,B}|_{r} + |f_{n−1,B}|_{r}) ≤ 4|e_{n−1,K}|_{r},
e_{n−1,Q} = (x_{2} + it_{1})ζ · (f_{n−1,B}(z_{1}, 0, x, t) − f_{n−1,B}(0, 0, x, t)) ⟹
|e_{n−1,Q}|_{r} ≤ 2r^{2}(|f_{n−1,B}|_{r} + |f_{n−1,B}|_{r}) ≤ 4|e_{n−1,K}|_{r},
e_{n−1,R} = x_{2}ζf_{n−1,B}(0, 0, x, t) ⟹
|e_{n−1,R}|_{r} ≤ r^{2}|f_{n−1,B}|_{r} ≤ |e_{n−1,K}|_{r},
e_{n−1,S} = (it_{1})ζ · (f_{n−1,B}(0, 0, x, t)
−f_{n−1,B}(0, 0, 0, 0, x_{4}, …, x_{m−1}, t)) ⟹
|e_{n−1,S}|_{r} ≤ r^{2}(|f_{n−1,B}|_{r} + |f_{n−1,B}|_{r}) ≤ 2|e_{n−1,K}|_{r},
e_{n−1,T} = (it_{1})ζ · (f_{n−1,B}(0, 0, 0, 0, x_{4}, …, x_{m−1}, t)
−f_{n−1,B}(0, 0, 0, t)) ⟹
|e_{n−1,T} |_{r} ≤ r^{2}(|f_{n−1,B}|_{r} + |f_{n−1,B}|_{r}) ≤ 2|e_{n−1,K}|_{r},
e_{n−1,U} = (it_{1})ζf_{n−1,B}(0, 0, 0, t) ⟹
|e_{n−1,U}|_{r} ≤ r^{2}|f_{n−1,B}|_{r} ≤ |e_{n−1,K}|_{r}.

We also make a more specific choice of normalization of p_{1}(z⃗, w⃗) as follows:

p_{1} = p_{1A} + p_{1B} + p_{1C} + p_{1D} + p_{1E} + p_{1F} + p_{1G}
p_{1A} = ∑_{α even, μ > 0} p_{1A}^{αλμJβK}z_{1}^{α}z_{2}^{λ}z_{3}^{μ}z^{J}z_{n−1}^{β}w^{K}
p_{1B} = ∑_{α even, α > 0} p_{1B}^{αλ0JβK}z_{1}^{α}z_{2}^{λ}z^{J}z_{n−1}^{β}w^{K}
p_{1C} = ∑_{β > 0} p_{1C}^{0λ0JβK}z_{2}^{λ}z^{J}z_{n−1}^{β}w^{K}
p_{1D} = ∑_{λ > 0} p_{1D}^{0λ0J0K}z_{2}^{λ}z^{J}w^{K}
p_{1E} = ∑_{J ≠ 0} p_{1E}^{000J0K}z^{J}w^{K}
p_{1F} = ∑ p_{1F}^{00000K}w^{K}
p_{1G} = ∑_{α odd} p_{1G}^{αλ0JβK}z_{1}^{α}z_{2}^{λ}z^{J}z_{n−1}^{β}w^{K}.

Now we can re-group the RHS of (90) to get the equation:

(95) 0 = e_{n−1,A} + p_{n−1}^{E} − 2z̄_{1}\overline{p_{1G}}
+e_{n−1,L} + e_{n−1,N} + (z_{1} + x_{2} + it_{1} + ix_{3}^{2})z̄_{1}p_{n−1}^{O}
+e_{n−1,D} − e_{n−1,M} − e_{n−1,O} − 2z̄_{1}\overline{p_{1A}}
+e_{n−1,E} − e_{n−1,P} − 2z̄_{1}\overline{p_{1B}}
+e_{n−1,F} − e_{n−1,Q} − 2z̄_{1}\overline{p_{1C}}
+e_{n−1,G} − e_{n−1,R} − e_{n−1,S} − 2z̄_{1}\overline{p_{1D}}
+e_{n−1,H} − e_{n−1,T} − 2z̄_{1}\overline{p_{1E}}
+e_{n−1,I} − e_{n−1,U} − 2z̄_{1}\overline{p_{1F}} ,

with like terms in each line, so that setting each line equal to zero gives, by inspection, a unique solution for p_{n−1}^{O} and p_{1A}, …, p_{1F}. The quantity p_{1G} will be found later and then p_{n−1}^{E} can be determined.

We record the estimates for the solved terms, beginning with p_{n−1}^{O}:

p_{n−1}^{O}(z_{1}, x, z̄_{1}^{2}, t) = −\frac{e_{n−1,L} + e_{n−1,N}}{(z_{1} + x_{2} + it_{1} + ix_{3}^{2})z̄_{1}}
= −\frac{e_{n−1,J} + e_{n−1,K}}{z_{1}z̄_{1}} = −\frac{e_{n−1,C}}{z_{1}z̄_{1}}
⟹ p_{n−1}^{O}(z_{1}, x, ζ^{2}, t) = −\frac{e_{n−1,C}(z_{1}, ζ, x, t)}{z_{1}ζ}
⟹ ||p_{n−1}^{O}||_{r} = sup_{|z_{1}|<r,|x_{σ}|<r,|ζ^{2}|<r^{2},|t_{α}|<r} |p_{n−1}^{O}(z_{1}, x, ζ^{2}, t)|

= sup_{|z_{1}|<r,|x_{σ}|<r,|ζ|<r,|t_{α}|<r} |−\frac{e_{n−1,C}(z_{1}, ζ, x, t)}{z_{1}ζ}|
(96)  ≤ \frac{1}{r^{2}} |e_{n−1,C}|_{r}
⟹ ∥z_{n}p_{n−1}^{O}∥_{r} ≤ ∥z_{n}∥_{r}∥p_{n−1}^{O}∥_{r} ≤ (3 + r)|e_{n−1}|_{r}.

The above step (96) uses the Schwarz Lemma twice, in steps similar to (94). The following calculation uses the Schwarz Lemma again, and also Lemma 7.5, and later calculations with similar steps may use these Lemmas without mention.

p_{1A}(z_{1}, x, z̄_{1}^{2}, t) = \overline{(\frac{e_{n−1,D} − e_{n−1,M} − e_{n−1,O}}{2z̄_{1}} )}
⟹ p_{1A}(z_{1}, x, ζ^{2}, t) = \frac{e′_{n−1,D} − e′_{n−1,M} − e′_{n−1,O}}{2z_{1}}
⟹ ∥p_{1A}∥_{r} ≤ \frac{1}{2r}(|e′_{n−1,D}|_{r} + |e′_{n−1,M}|_{r} + |e′_{n−1,O}|_{r})
≤ \frac{1}{2r}(2|e_{n−1,B}|_{r} + (2 + r)|e_{n−1,J}|_{r} + r|e_{n−1,K}|_{r})
≤ (\frac{3}{r} + \frac{3}{2})|e_{n−1}|_{r}.
p_{1B}(z_{1}, x, z̄_{1}^{2}, t) = \overline{(\frac{e_{n−1,E} − e_{n−1,P}}{2z̄_{1}} )}
⟹ p_{1B}(z_{1}, x, ζ^{2}, t) = \frac{e′_{n−1,E} − e′_{n−1,P}}{2z_{1}}
⟹ ∥p_{1B}∥_{r} ≤ \frac{1}{2r}(|e′_{n−1,E}|_{r} + |e′_{n−1,P} |_{r})
≤ \frac{1}{2r}(2|e_{n−1,B}|_{r} + 4|e_{n−1,K}|_{r}) ≤ \frac{3}{r}|e_{n−1}|_{r}.
p_{1C}(z_{1}, x, z̄_{1}^{2}, t) = \overline{(\frac{e_{n−1,F} − e_{n−1,Q}}{2z̄_{1}} )}
⟹ p_{1C}(z_{1}, x, ζ^{2}, t) = \frac{e′_{n−1,F} − e′_{n−1,Q}}{2z_{1}}
⟹ ∥p_{1C}∥_{r} ≤ \frac{1}{2r}(|e′_{n−1,F} |_{r} + |e′_{n−1,Q}|_{r})
≤ \frac{1}{2r}(2|e_{n−1,B}|_{r} + 4|e_{n−1,K}|_{r}) ≤ \frac{3}{r}|e_{n−1}|_{r}.
p_{1D}(z_{1}, x, z̄_{1}^{2}, t) = \overline{(\frac{e_{n−1,G} − e_{n−1,R} − e_{n−1,S}}{2z̄_{1}} )}
⟹ p_{1D}(z_{1}, x, ζ^{2}, t) = \frac{e′_{n−1,G} − e′_{n−1,R} − e′_{n−1,S}}{2z_{1}}
⟹ ∥p_{1D}∥_{r} ≤ \frac{1}{2r}(|e′_{n−1,G}|_{r} + |e′_{n−1,R}|_{r} + |e′_{n−1,S}|_{r})
≤ \frac{1}{2r}(2|e_{n−1,B}|_{r} + |e_{n−1,K}|_{r} + 2|e_{n−1,K}|_{r})
≤ \frac{5}{2r}|e_{n−1}|_{r}.

p_{1E}(z_{1}, x, z̄_{1}^{2}, t) = \overline{(\frac{e_{n−1,H} − e_{n−1,T}}{2z̄_{1}} )}
⟹ p_{1E}(z_{1}, x, ζ^{2}, t) = \frac{e′_{n−1,H} − e′_{n−1,T}}{2z_{1}}
⟹ ∥p_{1E}∥_{r} ≤ \frac{1}{2r}(|e′_{n−1,H}|_{r} + |e′_{n−1,T} |_{r})
≤ \frac{1}{2r}(2|e_{n−1,B}|_{r} + 2|e_{n−1,K}|_{r}) ≤ \frac{2}{r}|e_{n−1}|_{r}.
p_{1F} (z_{1}, x, z̄_{1}^{2}, t) = \overline{(\frac{e_{n−1,I} − e_{n−1,U}}{2z̄_{1}} )}
⟹ p_{1F} (z_{1}, x, ζ^{2}, t) = \frac{e′_{n−1,I} − e′_{n−1,U}}{2z_{1}}
⟹ ∥p_{1F} ∥_{r} ≤ \frac{1}{2r}(|e′_{n−1,I}|_{r} + |e′_{n−1,U}|_{r})
≤ \frac{1}{2r}(|e_{n−1,B}|_{r} + |e_{n−1,K}|_{r}) ≤ \frac{1}{r}|e_{n−1}|_{r}.

The norm of p_{1} cannot be estimated until we find p_{1G}, but the other terms can be combined to get:

p_{1A} + … + p_{1F} (97)

= \frac{1}{2z_{1}} (e′_{n−1,D} + … + e′_{n−1,I} − e′_{n−1,M} − (e′_{n−1,O} + … + e′_{n−1,U}))
= \frac{1}{2z_{1}} (e′_{n−1,B} − e′_{n−1,M} − (\frac{x_{2} + it_{1} + ix_{3}^{2}}{z_{1}} e_{n−1,K})′)
⟹ ∥p_{1A} + … + p_{1F} ∥_{r}
≤ \frac{1}{2r}|e_{n−1}|_{r} + \frac{1}{2r}(2 + r)2|e_{n−1}|_{r} + \frac{1}{2r} \frac{2r + r^{2}}{r} |e_{n−1}|_{r}
= (\frac{7}{2r} + \frac{3}{2})|e_{n−1}|_{r}.

The next step is to consider Equation (87), and to solve for the unknowns p_{n}, P_{1}, p_{1G}, p_{2}, and p_{3}. The unknown quantities p_{2}, p_{3} also appear in the σ = 2, 3 cases of (84), so we will need to consider a system of three equations:

0 = E_{2}(z_{1}, z̄_{1}, x, t) + Im(p_{2}(z⃗, t))
0 = E_{3}(z_{1}, z̄_{1}, x, t) + Im(p_{3}(z⃗, t))
(98) 0 = e_{n}(z_{1}, z̄_{1}, x, t) + p_{n}^{E}(z_{1}, x, z̄_{1}^{2}, t)
+(z_{1} + x_{2} + it_{1} + ix_{3}^{2})z̄_{1}p_{n}^{O}(z_{1}, x, z̄_{1}^{2}, t) + p_{n}^{L}(z⃗, t)

−z̄_{1} · (p_{1}(z⃗, t) + Re(p_{2}(z⃗, t)) + iP_{1}(t) + 2ix_{3}Re(p_{3}(z⃗, t)))

−\overline{(p_{1}(z⃗, t))} · (z_{1} + x_{2} + it_{1} + ix_{3}^{2}).

Finding the term p_{1G} will finish the solution for p_{1} and then p_{n−1}^{E} can be determined in (95).

Starting with the RHS of (84), the following decomposition of E_{σ} will be used for all σ = 2, …, m − 1:

E_{σ} = E_{σ,A} + e_{σ,B} + \overline{e_{σ,B}} + e_{σ,C} + \overline{e_{σ,C}} + E_{σ,D},
E_{σ,A} = ∑_{a, b even} E_{σ}^{abIK}z_{1}^{a}z̄_{1}^{b}x^{I}t^{K}
e_{σ,B} = ∑_{a > b, a odd, b even} E_{σ}^{abIK}z_{1}^{a}z̄_{1}^{b}x^{I}t^{K}
e_{σ,C} = ∑_{a > b, a even, b odd} E_{σ}^{abIK}z_{1}^{a}z̄_{1}^{b}x^{I}t^{K}
E_{σ,D} = ∑_{a, b odd} E_{σ}^{abIK}z_{1}^{a}z̄_{1}^{b}x^{I}t^{K}
= e_{σ,E} − e_{σ,F} ,
e_{σ,E} = (z_{1} + x_{2} + it_{1} + ix_{3}^{2})z̄_{1} ∑_{a, b odd} E_{σ}^{abIK}z_{1}^{a−1}z̄_{1}^{b−1}x^{I}t^{K}
e_{σ,F} = (x_{2} + it_{1} + ix_{3}^{2}) ∑_{a, b odd} E_{σ}^{abIK}z_{1}^{a−1}z̄_{1}^{b}x^{I}t^{K}.

The E_{σ,A} piece is simply an even part, so |E_{σ,A}|_{r} ≤ |E_{σ}|_{r}, and similarly for the odd part, |E_{σ,D}|_{r} ≤ |E_{σ}|_{r}. The other two subseries satisfy the estimate from Lemma 7.3: |e_{σ,B}|_{R} ≤ \frac{r^{2}}{(r−R)^{2}} |E_{σ}|_{r} and |e_{σ,C}|_{R} ≤ \frac{r^{2}}{(r−R)^{2}} |E_{σ}|_{r}. We re-group some of these subseries:

m_{σ}(z_{1}, z̄_{1}, x) = \overline{e_{σ,B}} + e_{σ,C} − e_{σ,F} = ∑_{a even, b odd} m_{σ}^{abIK}z_{1}^{a}z̄_{1}^{b}x^{I}t^{K}
f_{σ}(z_{1}, z̄_{1}, x) = E_{σ,A} + e_{σ,B} + \overline{e_{σ,C}} + \overline{m_{σ}} = ∑_{b even} f_{σ}^{abIK}z_{1}^{a}z̄_{1}^{b}x^{I}t^{K},

So, for each σ, there is a decomposition of the real valued series E_{σ} of the form

(99)  E_{σ} = f_{σ} + e_{σ,E} + m_{σ} − \overline{m_{σ}},

where f_{σ} is even in z̄_{1} and m_{σ} is even in z_{1} and odd in z̄_{1}. This decomposition is similar to that of E_{4} in [C_{6}].

The estimates follow from Lemma 7.5 and the Schwarz Lemma:

|e_{σ,E}|_{r} = |\frac{z_{1} + x_{2} + it_{1} + ix_{3}^{2}}{z_{1}} E_{σ,D}|_{r}
≤ (3 + r)|E_{σ}|_{r},
|m_{σ}|_{R} = |e′_{σ,B} + e_{σ,C} − e_{σ,F} |_{R} ≤ |e_{σ,B}|_{R} + |e_{σ,C}|_{R} + |e_{σ,F} |_{R}
≤ 2\frac{r^{2}}{(r − R)^{2}} |E_{σ}|_{r} + |x_{2} + it_{1} + ix_{3}^{2}|_{R} |\frac{E_{σ,D}}{z_{1}} |_{R}
≤ (\frac{2r^{2}}{(r − R)^{2}} + 2 + R) |E_{σ}|_{r},
|f_{σ}|_{R} = |E_{σ,A} + 2e_{σ,B} + 2e′_{σ,C} − e′_{σ,F} |_{R}
≤ (\frac{4r^{2}}{(r − R)^{2}} + 3 + R) |E_{σ}|_{r}.

For σ = 2 only, we decompose m_{2} = m_{2A} + m_{2B}:

m_{2A}(z_{1}, z̄_{1}, x, t) = ∑_{a even, b odd, d > 0} m_{2}^{abcdJK}z_{1}^{a}z̄_{1}^{b}x_{2}^{c}x_{3}^{d}x^{J}t^{K}
m_{2A}(z_{1}, ζ, x, t) = m_{2}(z_{1}, ζ, x_{2}, x_{3}, x_{4}, …, x_{m−1}, t)
−m_{2}(z_{1}, ζ, x_{2}, 0, x_{4}, …, x_{m−1}, t)
⟹ |m_{2A}|_{R} ≤ 2|m_{2}|_{R},
m_{2B}(z_{1}, z̄_{1}, x, t) = ∑_{a even, b odd} m_{2}^{abc0JK}z_{1}^{a}z̄_{1}^{b}x_{2}^{c}x^{J}t^{K},
m_{2B}(z_{1}, ζ, x, t) = m_{2}(z_{1}, ζ, x_{2}, 0, x_{4}, …, x_{m−1}, t)
⟹ |m_{2B}|_{R} ≤ |m_{2}|_{R}.

Recalling that the quantities p_{1A}, …, p_{1F} have already been determined in terms of e_{n−1}, we re-write Equation (98) with the known terms first (100–101), and introduce some new known quantities by adding and subtracting (102–103) some of the terms from the decomposition of E_{2} and E_{3}:

(100)0 = e_{n}(z_{1}, z̄_{1}, x, t) − z̄_{1} · (p_{1A} + … + p_{1F})

−(z_{1} + x_{2} + it_{1} + ix_{3}^{2}) · \overline{(p_{1A} + … + p_{1F})} (101) +z̄_{1} · (im_{2} − i\overline{m_{2}} − 2x_{3}m_{3} + 2x_{3}\overline{m_{3}}) (102) −iz̄_{1}m_{2} + iz̄_{1}\overline{m_{2A}} + iz̄_{1}\overline{m_{2B}} + 2z̄_{1}x_{3}m_{3} − 2z̄_{1}x_{3}\overline{m_{3}} (103)

+p_{n}^{E}(z_{1}, x, z̄_{1}^{2}, t)
+(z_{1} + x_{2} + it_{1} + ix_{3}^{2})z̄_{1}p_{n}^{O}(z_{1}, x, z̄_{1}^{2}, t) + p_{n}^{L}(z⃗, t)
−z̄_{1} · (p_{1G}(z⃗, t) + Re(p_{2}(z⃗, t)) + iP_{1}(t) + 2ix_{3}Re(p_{3}(z⃗, t)))
−\overline{(p_{1G}(z⃗, t))} · (z_{1} + x_{2} + it_{1} + ix_{3}^{2}).

An analogous add-and-subtract trick was used in [C_{6}]; here it will eventually lead to a cancellation that allows for the solution of p_{2} and p_{3} in terms of e_{n}, e_{n−1}, E_{2}, and E_{3}.

To get all the unknown terms to correspond to the given terms, we continue with a series decomposition of e_{n}:

As in (91), decompose e_{n} into even and odd parts e_{n,A}, e_{n,B}, e_{n,C}:

e_{n} = ∑ e_{n}^{abIK}z_{1}^{a}z̄_{1}^{b}x^{I}t^{K} = e_{n,A} + e_{n,B} + e_{n,C},
e_{n,A} = ∑_{b even} e_{n}^{abIK}z_{1}^{a}z̄_{1}^{b}x^{I}t^{K}
e_{n,B} = ∑_{a even, b odd} e_{n}^{abIK}z_{1}^{a}z̄_{1}^{b}x^{I}t^{K}
e_{n,C} = ∑_{a, b odd} e_{n}^{abIK}z_{1}^{a}z̄_{1}^{b}x^{I}t^{K}.

As with e_{n−1}, each of these subseries has norm bounded by |e_{n}|_{r}. The e_{n,B} subseries is further decomposed:

e_{n,B} = e_{n,D} + e_{n,E} + e_{n,F} + e_{n,G} + e_{n,H} + e_{n,I},
e_{n,D} = ∑_{a even, b odd, d > 0} e_{n}^{abcdJK}z_{1}^{a}z̄_{1}^{b}x_{2}^{c}x_{3}^{d}x^{J}t^{K}
e_{n,E} = ∑_{a even, b odd, a > 0} e_{n}^{abc0JK}z_{1}^{a}z̄_{1}^{b}x_{2}^{c}x^{J}t^{K}
e_{n,F} = ∑_{b odd, b > 1} e_{n}^{0bc0JK}z̄_{1}^{b}x_{2}^{c}x^{J}t^{K}
e_{n,G} = ∑_{c > 0} e_{n}^{01c0JK}z̄_{1}x_{2}^{c}x^{J}t^{K}
e_{n,H} = ∑_{J ≠ 0} e_{n}^{0100JK}z̄_{1}x^{J}t^{K}
e_{n,I} = ∑ e_{n}^{01000K}z̄_{1}t^{K}.

This is not exactly the same as the e_{n−1,B} decomposition (92); the e_{n,E} and e_{n,F} terms are defined differently. However, the estimates are similar: |e_{n,D}|_{r} ≤ 2|e_{n,B}|_{r} ≤ 2|e_{n}|_{r}, |e_{n,E}|_{r} ≤ 2|e_{n,B}|_{r} ≤ 2|e_{n}|_{r}, and using the Schwarz Lemma, |e_{n,F} |_{r} ≤ 2|e_{n,B}|_{r} ≤ 2|e_{n}|_{r}, |e_{n,G}|_{r} ≤ 2|e_{n,B}|_{r} ≤ 2|e_{n}|_{r}, |e_{n,H}|_{r} ≤ 2|e_{n,B}|_{r} ≤ 2|e_{n}|_{r}, and |e_{n,I}| ≤ |e_{n,B}|_{r} ≤ |e_{n}|_{r}.

The e_{n,C} subseries also decomposes into parts:

e_{n,C} = e_{n,J} + e_{n,K},
e_{n,J} = ∑_{a, b odd, d > 0} e_{n}^{abcdJK}z_{1}^{a}z̄_{1}^{b}x_{2}^{c}x_{3}^{d}x^{J}t^{K}
e_{n,K} = ∑_{a, b odd} e_{n}^{abc0JK}z_{1}^{a}z̄_{1}^{b}x_{2}^{c}x^{J}t^{K},

and this is just like the e_{n−1,C} decomposition (93), with similar estimates: |e_{n,J}|_{r} ≤ 2|e_{n,C}|_{r} ≤ 2|e_{n}|_{r}, |e_{n,K}|_{r} ≤ |e_{n,C}|_{r} ≤ |e_{n}|_{r}.

82 UNFOLDING CR SINGULARITIES

To start collecting some like terms in the known quantities, let:

f_{n,A} = e_{n,J} + iz̄_{1}\overline{m_{2A}} − 2z̄_{1}x_{3}\overline{m_{3}}
= ∑_{a, b odd, d > 0} f_{n,A}^{abcdJK}z_{1}^{a}z̄_{1}^{b}x_{2}^{c}x_{3}^{d}x^{J}t^{K} = f_{n,B} − f_{n,C},
f_{n,B} = ∑_{a, b odd, d > 0} f_{n,A}^{abcdJK}z_{1}^{a−1}z̄_{1}^{b−1}x_{2}^{c}x_{3}^{d}x^{J}t^{K} · (z_{1} + x_{2} + it_{1} + ix_{3}^{2})z̄_{1}
f_{n,C} = ∑_{a, b odd, d > 0} f_{n,A}^{abcdJK}z_{1}^{a−1}z̄_{1}^{b}x_{2}^{c}x_{3}^{d}x^{J}t^{K} · (x_{2} + it_{1} + ix_{3}^{2}).

Let:

f_{n,E}(z̄_{1}, x, t) = e_{n,F} − z̄_{1}p_{1C} = ∑_{b odd, b > 1} f_{n,E}^{0bc0JK}z̄_{1}^{b}x_{2}^{c}x^{J}t^{K},

and let g_{n,A}(ζ, x, t) = \frac{1}{ζ}f_{n,E}(ζ, x, t), so that

f_{n,E}(z̄_{1}, x, t) = z̄_{1} · (g_{n,A} + \overline{g_{n,A}}) − z̄_{1}\overline{g_{n,A}}.

Let:

f_{n,D} = e_{n,E} − z̄_{1}p_{1B} − z̄_{1}\overline{g_{n,A}}
= ∑_{a even, b odd, a > 0} f_{n,D}^{abc0JK}z_{1}^{a}z̄_{1}^{b}x_{2}^{c}x^{J}t^{K} = f_{n,F} − f_{n,G} − f_{n,H},
f_{n,F} = ∑_{a even, b odd, a > 0}f_{n,D}^{abc0JK}z_{1}^{a−1}z̄_{1}^{b−1}x_{2}^{c}x^{J}t^{K}·(z_{1} + x_{2} + it_{1} + ix_{3}^{2})z̄_{1}
f_{n,G} = ∑_{a even, b odd, a > 0} f_{n,D}^{abc0JK}z_{1}^{a−1}z̄_{1}^{b}x_{2}^{c}x^{J}t^{K} · (x_{2} + it_{1})
f_{n,H} = ∑_{a even, b odd, a > 0} f_{n,D}^{abc0JK}z_{1}^{a−1}z̄_{1}^{b}x_{2}^{c}x_{3}^{d}x^{J}t^{K} · (ix_{3}^{2}) = f_{n,I} − f_{n,J},
f_{n,I} =  ∑_{a even, b odd, a > 0} f_{n,D}^{abc0JK}z_{1}^{a−2}z̄_{1}^{b−1}x_{2}^{c}x_{3}^{d}x^{J}t^{K}·(ix_{3}^{2})·(z_{1} + x_{2} + it_{1} + ix_{3}^{2})z̄_{1}
f_{n,J} = ∑_{a even, b odd, a > 0} f_{n,D}^{abc0JK}z_{1}^{a−2}z̄_{1}^{b}x_{2}^{c}x_{3}^{d}x^{J}t^{K}·(ix_{3}^{2})·(x_{2} + it_{1} + ix_{3}^{2}),

so f_{n,D} = f_{n,F} − f_{n,G} − f_{n,I} + f_{n,J}.

Using the Schwarz Lemma and Lemma 7.5 we get the estimate:

f_{n,G}(z_{1}, ζ, x, t) = \frac{(e_{n,E} − ζp_{1B} − ζg′_{n,A}) · (x_{2} + it_{1})}{z_{1}}
= (\frac{e_{n,E}}{z_{1}} − \frac{ζp_{1B}}{z_{1}} − \frac{ζe′_{n,F}}{z_{1}^{2}} + \frac{ζp′_{1C}}{z_{1}} ) · (x_{2} + it_{1})
⟹ |f_{n,G}|_{r} ≤ 2|e_{n,E}|_{r} + 2r|p_{1B}|_{r} + 2|e_{n,F} |_{r} + 2r|p_{1C}|_{r}
≤ 4|e_{n}|_{r} + 12|e_{n−1}|_{r}.

Let:

f_{n,K}(z̄_{1}, x, t) = e_{n,G} − z̄_{1}p_{1D} = ∑_{c > 0} f_{n,K}^{01c0JK}z̄_{1}x_{2}^{c}x^{J}t^{K},

and let g_{n,B}(x, t) = \frac{1}{ζ}f_{n,K}(ζ, x, t), so that

f_{n,K}(z̄_{1}, x, t) = \frac{1}{2}z̄_{1} · (g_{n,B} + \overline{g_{n,B}}) + \frac{1}{2}z̄_{1} · (g_{n,B} − \overline{g_{n,B}}).

Further, re-arrange the second quantity:

\frac{1}{2}z̄_{1} · (g_{n,B} − \overline{g_{n,B}})

= ∑_{c > 0} iIm(f_{n,K}^{01c0JK})z̄_{1}x_{2}^{c}x^{J}t^{K} = g_{n,C} − g_{n,D} − g_{n,E},
g_{n,C} = ∑_{c > 0} iIm(f_{n,K}^{01c0JK})x_{2}^{c−1}x^{J}t^{K} · (z_{1} + x_{2} + it_{1} + ix_{3}^{2})z̄_{1}
g_{n,D} = ∑_{c > 0} iIm(f_{n,K}^{01c0JK})x_{2}^{c−1}x^{J}t^{K}z_{1}z̄_{1}
g_{n,E} = ∑_{c > 0} iIm(f_{n,K}^{01c0JK})x_{2}^{c−1}x^{J}t^{K} · (it_{1} + ix_{3}^{2})z̄_{1}.

Note that this last re-arrangement is different from that of e_{n−1,J}, e_{n−1,K}, E_{σ,D}, f_{n,A}, f_{n,D}, and f_{n,H}. (A similar step appeared in [C_{5}].)

Using the Schwarz Lemma and Lemma 7.5 we get the estimate:

g_{n,D}(z_{1}, ζ, x, t) = \frac{ζ}{2} · (g_{n,B} − g′_{n,B}) · \frac{z_{1}}{x_{2}}
= \frac{z_{1}e_{n,G} − z_{1}ζp_{1D} − ζe′_{n,G} + z_{1}ζp′_{1D}}{2x_{2}}
⟹ |g_{n,D}|_{r} ≤ |e_{n,G}|_{r} + r|p_{1D}|_{r} ≤ 2|e_{n}|_{r} + \frac{5}{2}|e_{n−1}|_{r}.

Let:

f_{n,L}(z̄_{1}, x_{4}, …, x_{m−1}, t) = e_{n,H} − z̄_{1}p_{1E} = ∑_{J ≠ 0} f_{n,L}^{0100JK}z̄_{1}x^{J}t^{K}.

Recall that in the (m, n) = (4, 5) case, there are no such terms. For m > 4, in a manner analogous to (66), this can be re-arranged as:

f_{n,L} = ∑_{ℓ=4}^{m−1} f_{n,L,ℓ}(z̄_{1}, x, t),
f_{n,L,4} = ∑_{j_{4}>0} f_{n,L}^{0100JK}z̄_{1}x_{4}^{j_{4}}x_{5}^{j_{5}} ⋯ x_{m−1}^{j_{m−1}}t^{K}
= f_{n,L}(z̄_{1}, x_{4}, …, x_{m−1}, t) − f_{n,L}(z̄_{1}, 0, x_{5}, …, x_{m−1}, t), …,
f_{n,L,ℓ} = ∑_{j_{ℓ}>0} f_{n,L}^{0100JK}z̄_{1}x_{ℓ}^{j_{ℓ}} ⋯ x_{m−1}^{j_{m−1}}t^{K}
= f_{n,L}(z̄_{1}, 0, …, 0, x_{ℓ}, …, x_{m−1}, t)
−f_{n,L}(z̄_{1}, 0, …, 0, 0, x_{ℓ+1}, …, x_{m−1}, t), …,
f_{n,L,m−1} = ∑_{j_{m−1}>0} f_{n,L}^{0100JK}z̄_{1}x_{m−1}^{j_{m−1}}t^{K}
= f_{n,L}(z̄_{1}, 0, …, 0, x_{m−1}, t).

For each ℓ,

|f_{n,L,ℓ}(ζ, x, t)|_{r} ≤ 2|f_{n,L}|_{r} = 2|e_{n,H} − ζp_{1E}|_{r}
≤ 2(2|e_{n,B}|_{r} + r · \frac{2}{r}|e_{n−1}|_{r}) ≤ 4|e_{n}|_{r} + 4|e_{n−1}|_{r}.

Of course, there is a smaller estimate for the last quantity, f_{n,L,m−1}, but it will be simpler later to treat them all the same.

For ℓ = 4, …, m − 1, let g_{n,F,ℓ}(x, t) = \frac{1}{ζx_{ℓ}} f_{n,L,ℓ}(ζ, x, t), so that:

f_{n,L} = ∑_{j=2}^{\frac{m}{2} −1}(x_{2j}g_{n,F,2j}(x, t) + x_{2j+1}g_{n,F,2j+1}(x, t))z̄_{1}

= ∑_{j=2}^{\frac{m}{2} −1} ((Im(g_{n,F,2j+1}) + iIm(g_{n,F,2j})) · (x_{2j} + ix_{2j+1})z̄_{1}
+ (Re(g_{n,F,2j}) − Im(g_{n,F,2j+1})) x_{2j}z̄_{1}
+ (Re(g_{n,F,2j+1}) + Im(g_{n,F,2j})) x_{2j+1}z̄_{1}) .

Finally, let:

f_{n,M}(z̄_{1}, t) = e_{n,I} − z̄_{1}p_{1F} = ∑ f_{n,M}^{01000K}z̄_{1}t^{K},

and let g_{n,G}(t) = \frac{1}{ζ}f_{n,M}(ζ, t), so that

f_{n,M}(z̄_{1}, t) = \frac{1}{2}z̄_{1} · (g_{n,G} + \overline{g_{n,G}}) + \frac{1}{2}z̄_{1} · (g_{n,G} − \overline{g_{n,G}}).

To solve (100–103), it is sufficient to collect together groups of like terms from RHS, known and unknown, and to set each group equal to 0, so that we can solve for the unknowns one at a time. Grouping together quantities with even powers of z̄_{1} gives:

0 = e_{n,A} −iz̄_{1}m_{2} +2z̄_{1}x_{3}m_{3} −(z_{1} +x_{2} +it_{1} +ix_{3}^{2})·\overline{(p_{1A} + … + p_{1F})}+p_{n}^{E}.

This determines p_{n}^{E}, and we get the estimate

∥p_{n}^{E}∥_{R} ≤ |e_{n,A}|_{R} + R|m_{2}|_{R} + 2R^{2}|m_{3}|_{R}

+|(z_{1} + x_{2} + it_{1} + ix_{3}^{2}) · (p′_{1A} + … + p′_{1F})|_{R}

≤ |e_{n}|_{r} + (\frac{2r^{2}}{(r − R)^{2}} + 2 + R) (R|E_{2}|_{r} + 2R^{2}|E_{3}|_{r})

+(3 + R)(\frac{7}{2} + \frac{3}{2}R)|e_{n−1}|_{r}.

Collecting quantities which are odd in both z_{1} and z̄_{1}, and do not depend on x_{3},

0 = e_{n,K} + iz̄_{1}\overline{m_{2B}} − f_{n,G} − g_{n,D} − z̄_{1}p_{1G},

which determines p_{1G}:

p_{1G}(z_{1}, x, ζ^{2}, t) = \frac{e_{n,K} − f_{n,G} − g_{n,D}}{ζ} + im′_{2B}.

We get the estimate:

∥p_{1G}∥_{R} ≤ \frac{1}{r}|e_{n,K}|_{r} + \frac{1}{r}|f_{n,G}|_{r} + \frac{1}{r}|g_{n,D}|_{r} + |m′_{2B}|_{R} (104)
≤ \frac{7}{r}|e_{n}|_{r} + \frac{29}{2r}|e_{n−1}|_{r} + (\frac{2r^{2}}{(r − R)^{2}} + 2 + R) |E_{2}|_{r}.

Collecting quantities with (z_{1} + x_{2} + it_{1} + ix_{3}^{2})z̄_{1} as a factor,

0 = f_{n,B} + f_{n,F} − f_{n,I} + g_{n,C} − \overline{p_{1G}} · (z_{1} + x_{2} + it_{1} + ix_{3}^{2})
+p_{n}^{O} · (z_{1} + x_{2} + it_{1} + ix_{3}^{2})z̄_{1}.

This determines p_{n}^{O},

p_{n}^{O}(z_{1}, x, ζ^{2}, t) = −\frac{f_{n,B} + f_{n,F} − f_{n,I} + g_{n,C}}{(z_{1} + x_{2} + it_{1} + ix_{3}^{2})ζ} + \frac{p′_{1G}}{ζ}
= −\frac{f_{n,A} + f_{n,D} − f_{n,H}}{z_{1}ζ} − \frac{g_{n,C}}{(z_{1} + x_{2} + it_{1} + ix_{3}^{2})ζ}

+\frac{e′_{n,K} − f′_{n,G} − g′_{n,D}}{z_{1}ζ} − \frac{im_{2B}}{ζ} .

Since

−\frac{g_{n,C}}{(z_{1} + x_{2} + it_{1} + ix_{3}^{2})ζ} = −\frac{g_{n,B} − g′_{n,B}}{2x_{2}} = \frac{g′_{n,D}}{z_{1}ζ} ,

these terms cancel in the above expression. (This minor simplification is not crucial to the Proof.) Expanding this expression, we get an estimate:

p_{n}^{O} = −\frac{e_{n,J}}{z_{1}ζ} − \frac{im′_{2A}}{z_{1}} + \frac{2x_{3}m′_{3}}{z_{1}}

−\frac{e_{n,E}}{z_{1}ζ} + \frac{p_{1B}}{z_{1}} + \frac{e′_{n,F}}{z_{1}^{2}} − \frac{p′_{1C}}{z_{1}}

+\frac{ix_{3}^{2}e_{n,E}}{z_{1}^{2}ζ} − \frac{ix_{3}^{2}p_{1B}}{z_{1}^{2}} − \frac{ix_{3}^{2}e′_{n,F}}{z_{1}^{3}} + \frac{ix_{3}^{2}p′_{1C}}{z_{1}^{2}}

+\frac{e′_{n,K}}{z_{1}ζ} − \frac{f′_{n,G}}{z_{1}ζ} − \frac{im_{2B}}{ζ}

⟹ ∥z_{n}p_{n}^{O}∥_{R} ≤ (3 + R) · [(11 + 4R)|e_{n}|_{r} + (18 + 6R)|e_{n−1}|_{r}

+ (\frac{2r^{2}}{(r − R)^{2}} + 2 + R) (3R|E_{2}|_{r} + 2R^{2}|E_{3}|_{r})].

Collecting quantities with (x_{2j} + ix_{2j+1})z̄_{1} as a factor,

(105) 0 = ∑_{j=2}^{\frac{m}{2} −1} ((Im(g_{n,F,2j+1}) + iIm(g_{n,F,2j})) · (x_{2j} + ix_{2j+1})z̄_{1})

+ ∑_{τ=m}^{n−2} p_{n,τ}^{L}(x, t) · (x_{2(τ−m+2)} + ix_{2(τ−m+2)+1})z̄_{1}.

The corresponding terms in the sums match, with j = τ − m + 2, again using the assumption m = \frac{2}{3}(n + 1), as in (68). This determines each

p_{n,τ}^{L} and p_{n}^{L} = ∑_{τ=m}^{n−2} z_{τ}p_{n,τ}^{L}(z, w⃗):

p_{n,τ}^{L}(x, t) = \frac{if_{n,L,2(τ−m+2)+1}}{2ζx_{2(τ−m+2)+1}} − \frac{if′_{n,L,2(τ−m+2)+1}}{2z_{1}x_{2(τ−m+2)+1}}

−\frac{f_{n,L,2(τ−m+2)}}{2ζx_{2(τ−m+2)}} + \frac{f′_{n,L,2(τ−m+2)}}{2z_{1}x_{2(τ−m+2)}}

∥z_{τ}p_{n,τ}^{L}∥_{r} ≤ 2r^{2} (2\frac{|f_{n,L,2(τ−m+2)+1}|_{r}}{2r^{2}} + 2\frac{|f_{n,L,2(τ−m+2)}|_{r}}{2r^{2}} )
≤ 16|e_{n}|_{r} + 16|e_{n−1}|_{r}
⟹ ∥p_{n}^{L}∥_{r} ≤ 8(m − 4)(|e_{n}|_{r} + |e_{n−1}|_{r}).

Collecting some terms linear in z̄_{1} and otherwise depending only on t,

(106)  0 = \frac{1}{2}z̄_{1} · (g_{n,G} − \overline{g_{n,G}}) − iz̄_{1}P_{1}(t).

This determines P_{1}(t), and the solution satisfies the reality condition:

P_{1}(t) = \frac{1}{2i}(g_{n,G} − \overline{g_{n,G}})
P_{1}(w⃗) = \frac{1}{2i} (\frac{e_{n,I}}{ζ} − p_{1F} − \frac{e′_{n,I}}{z_{1}} + p′_{1F} )
∥P_{1}∥_{r} ≤ \frac{1}{r}|e_{n}|_{r} + \frac{1}{r}|e_{n−1}|_{r}.

Collecting all the remaining terms,

0 = e_{n,D} − f_{n,C} + z̄_{1}·(g_{n,A} + \overline{g_{n,A}}) + f_{n,J} + \frac{1}{2}z̄_{1}·(g_{n,B} + \overline{g_{n,B}}) − g_{n,E}

+ ∑_{j=2}^{\frac{m}{2} −1} ((Re(g_{n,F,2j}) − Im(g_{n,F,2j+1})) x_{2j}z̄_{1}

+ (Re(g_{n,F,2j+1}) + Im(g_{n,F,2j})) x_{2j+1}z̄_{1})

+\frac{1}{2}z̄_{1} · (g_{n,G} + \overline{g_{n,G}}) + z̄_{1} · (im_{2} − i\overline{m_{2}} − 2x_{3}m_{3} + 2x_{3}\overline{m_{3}})

−z̄_{1} · (Re(p_{2}(z⃗, t)) + 2ix_{3}Re(p_{3}(z⃗, t))).

The only remaining unknowns are p_{2} and p_{3}, so we move them to the LHS and divide by z̄_{1}:

Re(p_{2}) + 2ix_{3}Re(p_{3}) = \frac{e_{n,D} − f_{n,C} + f_{n,J} − g_{n,E}}{z̄_{1}}

+g_{n,A} + \overline{g_{n,A}} + \frac{1}{2}(g_{n,B} + \overline{g_{n,B}}) (107)

+ ∑_{j=2}^{\frac{m}{2} −1} ((Re(g_{n,F,2j}) − Im(g_{n,F,2j+1})) x_{2j}

+ (Re(g_{n,F,2j+1}) + Im(g_{n,F,2j})) x_{2j+1})

+\frac{1}{2}(g_{n,G} + \overline{g_{n,G}})

+im_{2} − i\overline{m_{2}} − 2x_{3}m_{3} + 2x_{3}\overline{m_{3}}.

By construction, several of the above terms, including \frac{g_{n,E}}{z̄_{1}} , or pairs of terms, are real valued, and the rest are divisible by x_{3}, so setting the imaginary parts equal gives

(108) Re(p_{3}) = \frac{iIm (\frac{e_{n,D}−f_{n,C}+f_{n,J}}{z̄_{1}} ) − 2x_{3}m_{3} + 2x_{3}\overline{m_{3}}}{2ix_{3}}
= Im (\frac{e_{n,D} − f_{n,C} + f_{n,J}}{2z̄_{1}x_{3}} ) + im_{3} − i\overline{m_{3}}.

From (98) and (99), we have

Im(p_{3}) = −(f_{3} + e_{3,E} + m_{3} − \overline{m_{3}}),

and combining Re(p_{3}) + iIm(p_{3}), the m_{3} terms cancel — this cancellation is a key step, as mentioned previously when the m_{2}, m_{3} terms were introduced in (102–103):

p_{3} = \frac{e_{n,D}}{4iζx_{3}} − \frac{e′_{n,D}}{4iz_{1}x_{3}} − \frac{f_{n,C}}{4iζx_{3}} + \frac{f′_{n,C}}{4iz_{1}x_{3}} + \frac{f_{n,J}}{4iζx_{3}} − \frac{f′_{n,J}}{4iz_{1}x_{3}} − if_{3} − ie_{3,E}
= p_{3}^{E}(z_{1}, x, ζ^{2}, t) + (z_{1} + x_{2} + it_{1} + ix_{3}^{2})ζp_{3}^{O}(z_{1}, x, ζ^{2}, t).

We get estimates on some of these terms using the Schwarz Lemma, and on their conjugates by Lemma 7.5. The e_{3,E} term determines p_{3}^{O}, the rest determine p_{3}^{E}.

(109)  p_{3}^{O} = \frac{−ie_{3,E}}{(z_{1} + x_{2} + it_{1} + ix_{3}^{2})ζ} = \frac{−iE_{3,D}}{z_{1}ζ}
⟹ ∥z_{n}p_{3}^{O}∥_{r} ≤ (3 + r)|E_{3}|_{r}.

|\frac{e_{n,D}}{4iζx_{3}} |_{R} ≤ \frac{2|e_{n}|_{r}}{4R^{2}} ,

\frac{f_{n,C}}{4iζx_{3}} = \frac{\frac{x_{2}+it_{1}+ix_{3}^{2}}{z_{1}} f_{n,A}}{4iζx_{3}}
= (x_{2} + it_{1} + ix_{3}^{2}) · (\frac{e_{n,J}}{4iz_{1}ζx_{3}} + \frac{m′_{2A}}{4z_{1}x_{3}} − \frac{m′_{3}}{2iz_{1}} )
⟹ |\frac{f_{n,C}}{4iζx_{3}} |_{R} ≤ (2R + R^{2}) (\frac{2|e_{n}|_{r}}{4R^{3}}

+ (\frac{2r^{2}}{(r − R)^{2}} + 2 + R) (\frac{2|E_{2}|_{r}}{4R^{2}} + \frac{|E_{3}|_{r}}{2R} )) ,

\frac{f_{n,J}}{4iζx_{3}} = \frac{\frac{(x_{2}+it_{1}+ix_{3}^{2})}{z_{1}}f_{n,H}}{4iζx_{3}} = \frac{x_{2} + it_{1} + ix_{3}^{2}}{4iζz_{1}x_{3}} · \frac{ix_{3}^{2}}{z_{1}} f_{n,D}
= (x_{2} + it_{1} + ix_{3}^{2})x_{3} · (\frac{e_{n,E}}{4z_{1}^{2}ζ} − \frac{p_{1B}}{4z_{1}^{2}} − \frac{e′_{n,F}}{4z_{1}^{3}} + \frac{p′_{1C}}{4z_{1}^{2}} )
⟹ |\frac{f_{n,J}}{4iζx_{3}} |_{R} ≤ (2R^{2} + R^{3}) (\frac{|e_{n}|_{r}}{R^{3}} + \frac{\frac{6}{r}|e_{n−1}|_{r}}{4R^{2}} )
⟹ ∥p_{3}^{E}∥_{R} ≤ \frac{|e_{n}|_{r}}{R^{2}} + (2 + R)\frac{|e_{n}|_{r}}{R^{2}}

+(2 + R) (\frac{2r^{2}}{(r − R)^{2}} + 2 + R) (\frac{|E_{2}|_{r}}{R} + |E_{3}|_{r})

+(2 + R) (\frac{2|e_{n}|_{r}}{R} + \frac{3|e_{n−1}|_{r}}{r} )

+ (\frac{4r^{2}}{(r − R)^{2}} + 3 + R) |E_{3}|_{r}.

Returning to (107), setting the real parts of LHS and RHS equal gives

Re(p_{2}) = Re (\frac{e_{n,D} − f_{n,C} + f_{n,J}}{z̄_{1}} ) − \frac{g_{n,E}}{z̄_{1}}

+g_{n,A} + \overline{g_{n,A}} + \frac{1}{2}(g_{n,B} + \overline{g_{n,B}})

+ ∑_{j=2}^{\frac{m}{2} −1} ((Re(g_{n,F,2j}) − Im(g_{n,F,2j+1})) x_{2j}

+ (Re(g_{n,F,2j+1}) + Im(g_{n,F,2j})) x_{2j+1})

+\frac{1}{2}(g_{n,G} + \overline{g_{n,G}})

+im_{2} − i\overline{m_{2}}.

From (98) and (99), we have

Im(p_{2}) = −(f_{2} + e_{2,E} + m_{2} − \overline{m_{2}}), and combining Re(p_{2}) + iIm(p_{2}), the m_{2} terms cancel, as before with p_{3} and m_{3}.

p_{2} = \frac{e_{n,D}}{2ζ} + \frac{e′_{n,D}}{2z_{1}} − \frac{f_{n,C}}{2ζ} − \frac{f′_{n,C}}{2z_{1}} + \frac{f_{n,J}}{2ζ} + \frac{f′_{n,J}}{2z_{1}}

−\frac{g_{n,E}}{ζ} + g_{n,A} + g′_{n,A} + \frac{1}{2}(g_{n,B} + g′_{n,B})

+ ∑_{j=2}^{\frac{m}{2} −1} (\frac{f_{n,L,2j}}{2ζ} + \frac{f′_{n,L,2j}}{2z_{1}} − \frac{f_{n,L,2j+1}x_{2j}}{2iζx_{2j+1}} + \frac{f′_{n,L,2j+1}x_{2j}}{2iz_{1}x_{2j+1}}

+ \frac{f_{n,L,2j+1}}{2ζ} + \frac{f′_{n,L,2j+1}}{2z_{1}} + \frac{f_{n,L,2j}x_{2j+1}}{2iζx_{2j}} − \frac{f′_{n,L,2j}x_{2j+1}}{2iz_{1}x_{2j}} )

+\frac{1}{2}(g_{n,G} + g′_{n,G}) − if_{2} − ie_{2,E}

= p_{2}^{E}(z_{1}, x, ζ^{2}, t) + (z_{1} + x_{2} + it_{1} + ix_{3}^{2})ζp_{2}^{O}(z_{1}, x, ζ^{2}, t).

Again similarly to the above calculation for p_{3}, the e_{2,E} term determines p_{2}^{O} as in (109), the rest determine p_{2}^{E}.

p_{2}^{O} = \frac{−ie_{2,E}}{(z_{1} + x_{2} + it_{1} + ix_{3}^{2})ζ} = \frac{−iE_{2,D}}{z_{1}ζ}
⟹ ∥z_{n}p_{2}^{O}∥_{r} ≤ (3 + r)|E_{2}|_{r}.

The estimates for the e_{n,D}, f_{n,C}, f_{n,J}, f_{n,L,ℓ} and f_{2} terms are similar to some previous calculations.

\frac{g_{n,E}}{ζ} = (\frac{e_{n,G}}{2ζx_{2}} − \frac{p_{1D}}{2x_{2}} − \frac{e′_{n,G}}{2z_{1}x_{2}} + \frac{p′_{1D}}{2x_{2}} ) · (it_{1} + ix_{3}^{2})
⟹ |\frac{g_{n,E}}{ζ} |_{R} ≤ 2(\frac{1}{R} + 1)|e_{n}|_{r} + \frac{5}{2r}(1 + R)|e_{n−1}|_{r},

g_{n,A} = \frac{e_{n,F}}{ζ} − p_{1C} ⟹ |g_{n,A}|_{R} ≤ \frac{2|e_{n}|_{r}}{R} + \frac{3}{r}|e_{n−1}|_{r},
g_{n,B} = \frac{e_{n,G}}{ζ} − p_{1D} ⟹ |g_{n,B}|_{R} ≤ \frac{2|e_{n}|_{r}}{R} + \frac{5}{2r}|e_{n−1}|_{r},
g_{n,G} = \frac{e_{n,I}}{ζ} − p_{1F} ⟹ |g_{n,G}|_{R} ≤ \frac{|e_{n}|_{r}}{R} + \frac{1}{r}|e_{n−1}|_{r}
⟹ ∥p_{2}^{E}∥_{R} ≤ \frac{2|e_{n}|_{r}}{R^{2}} + (4 + 2R)\frac{|e_{n}|_{r}}{R}

+(4 + 2R) (\frac{2r^{2}}{(r − R)^{2}} + 2 + R) (|E_{2}|_{r} + R|E_{3}|_{r})

+(4R + 2R^{2}) (\frac{2|e_{n}|_{r}}{R} + \frac{3|e_{n−1}|_{r}}{r} )

+2(\frac{1}{R} + 1)|e_{n}|_{r} + \frac{5}{2r}(1 + R)|e_{n−1}|_{r}

+\frac{4}{R}|e_{n}|_{r} + \frac{6}{r}|e_{n−1}|_{r} + \frac{2}{R}|e_{n}|_{r} + \frac{5}{2r}|e_{n−1}|_{r}

+8(m − 4)(|e_{n}|_{r} + |e_{n−1}|_{r}) + \frac{1}{R}|e_{n}|_{r} + \frac{1}{r}|e_{n−1}|_{r}

+ (\frac{4r^{2}}{(r − R)^{2}} + 3 + R) |E_{2}|_{r}.

This completes the solution of the system of equations (98). From (97) and (104), we can conclude that p_{1} = p_{1A} + … + p_{1F} + p_{1G} has been determined in terms of e_{n−1}, e_{n}, and E_{2}, so that

∥p_{1}∥_{R} ≤ (\frac{18}{r} + \frac{3}{2}) |e_{n−1}|_{r} + \frac{7}{r}|e_{n}|_{r} + (\frac{2r^{2}}{(r − R)^{2}} + 2 + R) |E_{2}|_{r}.

From (95),

p_{n−1}^{E} = −e_{n−1,A} + 2z̄_{1}\overline{p_{1G}} ⟹
∥p_{n−1}^{E}∥_{R} ≤ 30|e_{n−1}|_{r} + 14|e_{n}|_{r} + 2R (\frac{2r^{2}}{(r − R)^{2}} + 2 + R) |E_{2}|_{r}.

If (m, n) = (4, 5), we are done, all the equations have been solved. Otherwise, the remaining equations from (84–88) can be solved in blocks of three. The details of the calculation are similar to those in Theorem 5.6 of [C_{6}], and simpler than the previous subsystem (98). For τ = m, …, n − 2, the three equations to be solved are:

(110) 0 = E_{2(τ−m+2)}(z_{1}, z̄_{1}, x, t) + Im(p_{2(τ−m+2)}(z⃗, t))
0 = E_{2(τ−m+2)+1}(z_{1}, z̄_{1}, x, t) + Im(p_{2(τ−m+2)+1}(z⃗, t))
(111) 0 = e_{τ}(z_{1}, z̄_{1}, x, t) + p_{τ}(z⃗, t)
−\overline{p_{1}(z⃗, t)} · (x_{2(τ−m+2)} + ix_{2(τ−m+2)+1})
−z̄_{1} · (Re(p_{2(τ−m+2)}(z⃗, t)) + iRe(p_{2(τ−m+2)+1}(z⃗, t))).

We start by collecting some known quantities. Let

f_{τ}(z_{1}, z̄_{1}, x, t) = e_{τ}(z_{1}, z̄_{1}, x, t) − \overline{p_{1}(z⃗, t)} · (x_{2(τ−m+2)} + ix_{2(τ−m+2)+1}),

so that

|f_{τ}|_{R} ≤ |e_{τ}|_{R} + 2R ((\frac{18}{r} + \frac{3}{2}) |e_{n−1}|_{r}

+\frac{7}{r}|e_{n}|_{r} + (\frac{2r^{2}}{(r − R)^{2}} + 2 + R) |E_{2}|_{r}) ,

and as in (91), decompose it into even and odd parts:

f_{τ} = ∑ f_{τ}^{abIK}z_{1}^{a}z̄_{1}^{b}x^{I}t^{K} = f_{τ,A} + f_{τ,B} + f_{τ,C},
f_{τ,A} = ∑_{b even} f_{τ}^{abIK}z_{1}^{a}z̄_{1}^{b}x^{I}t^{K}
f_{τ,B} = ∑_{a, b odd} f_{τ}^{abIK}z_{1}^{a}z̄_{1}^{b}x^{I}t^{K}
f_{τ,C} = ∑_{a even, b odd} f_{τ}^{abIK}z_{1}^{a}z̄_{1}^{b}x^{I}t^{K}.

The same bound holds for the even and odd parts: |f_{τ,A}|_{R} ≤ |f_{τ}|_{R}, |f_{τ,B}|_{R} ≤ |f_{τ}|_{R}, and |f_{τ,B}|_{R} ≤ |f_{τ}|_{R}.

Then, adding and subtracting some known quantities to (111), as in (102–103), and using the normalized form of p_{τ}, gives:

0 = f_{τ,A} + f_{τ,B} + f_{τ,C}
+p_{τ}^{E}(z_{1}, x, z̄_{1}^{2}, t) + (z_{1} + x_{2} + it_{1} + ix_{3}^{2})z̄_{1}p_{2}^{O}(z_{1}, x, z̄_{1}^{2}, t)

+z̄_{1} · (im_{2(τ−m+2)} − i\overline{m_{2(τ−m+2)}} − m_{2(τ−m+2)+1} + \overline{m_{2(τ−m+2)+1}}) −iz̄_{1}m_{2(τ−m+2)} + iz̄_{1}\overline{m_{2(τ−m+2)}} + z̄_{1}m_{2(τ−m+2)+1} − z̄_{1}\overline{m_{2(τ−m+2)+1}}

−z̄_{1} · (Re(p_{2(τ−m+2)}(z⃗, t)) + iRe(p_{2(τ−m+2)+1}(z⃗, t))).

7. RAPID CONVERGENCE PROOF OF THE MAIN THEOREM 93

Let:

g_{τ,A} = f_{τ,B} − iz̄_{1}\overline{m_{2(τ−m+2)}} + z̄_{1}\overline{m_{2(τ−m+2)+1}}
= ∑_{a, b odd} g_{τ,A}^{abIK}z_{1}^{a}z̄_{1}^{b}x^{I}t^{K} = g_{τ,B} − g_{τ,C},
g_{τ,B} = ∑_{a, b odd} g_{τ,A}^{abIK}z_{1}^{a−1}z̄_{1}^{b−1}x^{I}t^{K} · (z_{1} + x_{2} + it_{1} + ix_{3}^{2})z̄_{1}
g_{τ,C} = ∑_{a, b odd} g_{τ,A}^{abIK}z_{1}^{a−1}z̄_{1}^{b}x^{I}t^{K} · (x_{2} + it_{1} + ix_{3}^{2}),

so

|g_{τ,C}|_{R} = |\frac{x_{2} + it_{1} + ix_{3}^{2}}{z_{1}} g_{τ,A}|_{R}
≤ (2 + R)[|f_{τ}|_{R}

+R (\frac{2r^{2}}{(r − R)^{2}} + 2 + R) (|E_{2(τ−m+2)}|_{r} + |E_{2(τ−m+2)+1}|_{r})].

Then (111) can be solved by comparing like terms. Grouping together quantities with even powers of z̄_{1} gives:

0 = f_{τ,A} − iz̄_{1}m_{2(τ−m+2)} + z̄_{1}m_{2(τ−m+2)+1} + p_{τ}^{E},

which determines p_{τ}^{E}, and

∥p_{τ}^{E}∥_{R}

≤ |f_{τ}|_{R} + R (\frac{2r^{2}}{(r − R)^{2}} + 2 + R) (|E_{2(τ−m+2)}|_{r} + |E_{2(τ−m+2)+1}|_{r}).

Another group is

0 = g_{τ,B} + (z_{1} + x_{2} + it_{1} + ix_{3}^{2})z̄_{1}p_{τ}^{O}(z_{1}, x, z̄_{1}^{2}, t),

which determines p_{τ}^{O}, and

p_{τ}^{O} = \frac{−g_{τ,A}}{z_{1}z̄_{1}} = − (\frac{f_{τ,B}}{z_{1}z̄_{1}} − i\frac{\overline{m_{2(τ−m+2)}}}{z_{1}} + \frac{\overline{m_{2(τ−m+2)+1}}}{z_{1}} )
⟹ ∥z_{n}p_{τ}^{O}∥_{R}
≤ (3R^{2} + R^{3}) (\frac{1}{R^{2}} |f_{τ}|_{R}

+\frac{1}{R} (\frac{2r^{2}}{(r − R)^{2}} + 2 + R) (|E_{2(τ−m+2)}|_{r} + |E_{2(τ−m+2)+1}|_{r})).

The remaining terms are:

0 = f_{τ,C} − g_{τ,C}

+z̄_{1} · (im_{2(τ−m+2)} − i\overline{m_{2(τ−m+2)}} − m_{2(τ−m+2)+1} + \overline{m_{2(τ−m+2)+1}})

−z̄_{1} · (Re(p_{2(τ−m+2)}(z⃗, t)) + iRe(p_{2(τ−m+2)+1}(z⃗, t))).

Dividing by z̄_{1} and considering the real and imaginary parts,

Re(p_{2(τ−m+2)}(z⃗, t))
= \frac{f_{τ,C} − g_{τ,C}}{2z̄_{1}} + \frac{\overline{f_{τ,C}} − \overline{g_{τ,C}}}{2z_{1}} + im_{2(τ−m+2)} − i\overline{m_{2(τ−m+2)}},
Re(p_{2(τ−m+2)+1}(z⃗, t))
= \frac{f_{τ,C} − g_{τ,C}}{2iz̄_{1}} − \frac{\overline{f_{τ,C}} − \overline{g_{τ,C}}}{2iz_{1}} + im_{2(τ−m+2)+1} − i\overline{m_{2(τ−m+2)+1}}.

From (110) and (99), we have

Im(p_{2(τ−m+2)}(z⃗, t))
= −(f_{2(τ−m+2)} + e_{2(τ−m+2),E} + m_{2(τ−m+2)} − \overline{m_{2(τ−m+2)}}),
Im(p_{2(τ−m+2)+1}(z⃗, t))
= −(f_{2(τ−m+2)+1} + e_{2(τ−m+2)+1,E} + m_{2(τ−m+2)+1} − \overline{m_{2(τ−m+2)+1}}).

Combining the real and imaginary parts of p_{2(τ−m+2)} and p_{2(τ−m+2)+1}, the m terms cancel, as before with p_{2} and p_{3}, to give:

p_{2(τ−m+2)}
= p_{2(τ−m+2)}^{E}(z_{1}, x, ζ^{2}, t)
+(z_{1} + x_{2} + it_{1} + ix_{3}^{2})ζp_{2(τ−m+2)}^{O}(z_{1}, x, ζ^{2}, t)
= \frac{f_{τ,C} − g_{τ,C}}{2ζ} + \frac{f′_{τ,C} − g′_{τ,C}}{2z_{1}} − if_{2(τ−m+2)} − ie_{2(τ−m+2),E},
p_{2(τ−m+2)+1}
= p_{2(τ−m+2)+1}^{E}(z_{1}, x, ζ^{2}, t)
+(z_{1} + x_{2} + it_{1} + ix_{3}^{2})ζp_{2(τ−m+2)+1}^{O}(z_{1}, x, ζ^{2}, t)
= \frac{f_{τ,C} − g_{τ,C}}{2iζ} − \frac{f′_{τ,C} − g′_{τ,C}}{2iz_{1}} − if_{2(τ−m+2)+1} − ie_{2(τ−m+2)+1,E}.

Similar estimates hold in both cases, σ = 2(τ −m+2), 2(τ −m+2)+1; the e_{σ,E} term determines p_{σ}^{O}, and the rest of the terms determine p_{σ}^{E}. As in (109),

∥z_{n}p_{σ}^{O}∥_{r} ≤ (3 + r)|E_{σ}|_{r},

and

∥p_{σ}^{E}∥_{R} ≤ \frac{1}{R}|f_{τ}|_{r} + \frac{1}{R}|g_{τ,C}|_{R} + (\frac{4r^{2}}{(r − R)^{2}} + 3 + R) |E_{σ}|_{r}.

The claimed estimate from the statement of the Theorem follows; it is possible, but not necessary, to explicitly calculate the polynomial expressions C_{m}^{1} and C_{m}^{2}.

Remark 7.7. The Proof of the above Theorem can be modified to apply to the degree d = 3 case, as discussed in Remark 6.6, to establish the existence of a holomorphic coordinate change that eliminates all cubic terms depending on t in (75), leaving only the cubic term iz̄_{1}x_{3}^{2}, and possibly changing the O(4) terms. In the nonlinear system of equations (80–82), if d = 3, then the solution p_{3} may have linear terms of the form ∑ r_{3}^{α}t_{α}, as remarked in (83), or as follows from the construction of the above Proof, where terms of the form z̄_{1}x_{3}t_{α} in e_{n,D} could contribute such real valued linear terms to p_{3}, from (108). In this case, the term −iz̄_{1} · (Re(p_{3}(z⃗, t)))^{2} from line (81) can have degree 3 terms, and so it cannot be included among the higher-degree terms that are neglected to arrive at the linear system (84–88). The only needed modification to the above Proof is that the degree 3 part of the nonlinear, but purely imaginary, quantity −iz̄_{1} · (Re(p_{3}(z⃗, t)))^{2} is inserted into Equation (106), to get:

(112) 0 = \frac{1}{2}z̄_{1} · (g_{n,G} − \overline{g_{n,G}}) − iz̄_{1} · (∑ r_{3}^{α}t_{α})^{2} − iz̄_{1}P_{1}(t).

Then, while p_{3} eliminates terms in e_{n} of the form z̄_{1}x_{3}t_{α}, there still exists a weight 2 solution P_{1}(t) of Equation (112) that eliminates the terms of the form z̄_{1}t_{α}t_{β}.

Corollary 7.8. There is a constant c_{1} > 0 (depending only on m), such that for any p⃗ and e⃗ as in Theorem 7.6, and any radius lengths ρ, r with \frac{1}{2} < ρ < r ≤ 1, the following hold:

max\{∥P_{1}∥_{ρ}, ∥p_{1}∥_{ρ}, …, ∥p_{n}∥_{ρ}\} ≤ \frac{c_{1}|e⃗|_{r}}{(r − ρ)^{2}}

max_{j=1,...,n} \{∥\frac{dP_{1}}{dz_{j}} ∥_{ρ} + ∑_{ℓ=1}^{n} ∥\frac{dp_{ℓ}}{dz_{j}} ∥_{ρ}\} ≤ \frac{c_{1}|e⃗|_{r}}{(r − ρ)^{3}} ,

max_{α=1,...,k} \{∥\frac{dP_{1}}{dw_{α}} ∥_{ρ} + ∑_{ℓ=1}^{n} ∥\frac{dp_{ℓ}}{dw_{α}} ∥_{ρ}\} ≤ \frac{c_{1}|e⃗|_{r}}{(r − ρ)^{3}} .

Proof. Let R = \frac{1}{2}(ρ + r). The bound on each p_{j} follows from ∥p_{j}∥_{ρ} ≤ ∥p_{j}∥_{R} and the bounds from Theorem 7.6: using \frac{1}{2} < R < r ≤ 1 ≤ \frac{1}{r} < \frac{1}{R} < 2, the quantities C_{m}^{1} and C_{m}^{2} are bounded above by some positive constant (depending on m), and (r − R)^{−2} satisfies: 16 < \frac{1}{(r−R)^{2}} = \frac{4}{(r−ρ)^{2}} . The bounds for the derivatives of p_{k} follow from this consequence ([A]) of Cauchy’s estimate:

If 0 < R_{2} < R_{1} and f(ζ) is holomorphic and bounded by K for |ζ| < R_{1}, then \frac{df}{dζ} is bounded by \frac{K}{R_{1}−R_{2}} for |ζ| < R_{2}.

This fact can be applied with K = ∥p_{ℓ}∥_{R} or ∥P_{1}∥_{R}, and R_{1} − R_{2} = R − ρ = \frac{1}{2}(r − ρ) for the z_{1}, …, z_{m−1} and w_{1}, …, w_{k} derivatives, R_{1} − R_{2} = R^{2} − ρ^{2} > R − ρ = \frac{1}{2}(r − ρ) for the z_{n−1} derivatives, R_{1} − R_{2} = 2R^{2} − 2ρ^{2} > r − ρ for the z_{m}, …, z_{n−2} derivatives, and R_{1} − R_{2} = (3R^{2} + R^{3}) − (3ρ^{2} + ρ^{3}) > \frac{15}{8}(r − ρ) for the z_{n} derivatives.

Of course, some derivatives can be given sharper estimates, for example, dP_{1}/dz_{j} = 0 by construction, but it will be simpler later to treat them all the same.

The lower bound r > \frac{1}{2} was important for the previous Corollary, but it is not a significant a priori restriction on the manifold ; M. By a real rescaling:

(z⃗, w⃗) ↦ (113)

(a_{1}z_{1}, a_{1}z_{2}, \sqrt{a_{1}}z_{3}, a_{1}z_{4}, …, a_{1}z_{m−1}, (a_{1})^{2}z_{m}, …, (a_{1})^{2}z_{n}, a_{1} · w⃗)^{T},

a_{1} > 0 (a special case of (12)), the equations (75) can be assumed to define ; M for |z_{1}| < 1, |x_{σ}| < 1, |t_{α}| < 1, and, further, for any η_{0} > 0, there is a rescaling making |e⃗|_{1} < η_{0}.

7.3. The new defining equations and some estimates.

To get a solution of the nonlinear system of equations (80) by iterating the solution of the linear system, the rapid convergence technique will apply, closely following the methods used by [Moser] on a different CR singularity problem. Each step along the way to a proof of Main Theorem 6.5 is stated as a Theorem.

Substituting the linear equation’s normalized solution p⃗ from Theorem 7.6 into E_{2}, …, e_{n} in the RHS of the nonlinear equations (80–82) gives a quantity q⃗ depending on z_{1}, z̄_{1}, x, t. Equation (82) is satisfied exactly by the P_{1} constructed in Theorem 7.6, so for now, only (80–81), where the solution of the linear equations is merely an approximation to a solution of the nonlinear equations, will be considered. Let (114) z⃗ =(z_{1}, x, …, (x_{2(τ−m+2)}+ix_{2(τ−m+2)+1})z̄_{1}, …, z̄_{1}^{2}, (z_{1}+x_{2}+it_{1}+ix_{3}^{2})z̄_{1})^{T}, as in (88), let

(115) z⃗ + e⃗ = (z_{1}, x_{2} + iE_{2}, …, x_{m−1} + iE_{m−1},
…, (x_{2(τ−m+2)} + ix_{2(τ−m+2)+1})z̄_{1} + e_{τ}, …,
z̄_{1}^{2} + e_{n−1}, (z_{1} + x_{2} + it_{1} + ix_{3}^{2})z̄_{1} + e_{n})^{T} ,

as in (79), and then define q⃗(z_{1}, z̄_{1}, x, t) = (Q_{2}, …, Q_{m−1}, q_{m}, …, q_{n}) by:

(116) Q_{σ} = Im(p_{σ}(z⃗ + e⃗, t) − p_{σ}(z⃗, t))
(117) q_{τ} = p_{τ}(z⃗ + e⃗, t) − p_{τ}(z⃗, t)
−(x_{2(τ−m+2)} + ix_{2(τ−m+2)+1}) · \overline{(p_{1}(z⃗ + e⃗, t) − p_{1}(z⃗, t))}
−z̄_{1}Re(p_{2(τ−m+2)}(z⃗ + e⃗, t) − p_{2(τ−m+2)}(z⃗, t))
−iz̄_{1}Re(p_{2(τ−m+2)+1}(z⃗ + e⃗, t) − p_{2(τ−m+2)+1}(z⃗, t))
−\overline{p_{1}(z⃗ + e⃗, t)} · Re(p_{2(τ−m+2)}(z⃗ + e⃗, t))
−i\overline{p_{1}(z⃗ + e⃗, t)} · Re(p_{2(τ−m+2)+1}(z⃗ + e⃗, t))
(118) q_{n−1} = p_{n−1}(z⃗ + e⃗, t) − p_{n−1}(z⃗, t)
−2z̄_{1}\overline{(p_{1}(z⃗ + e⃗, t) − p_{1}(z⃗, t))} − \overline{(p_{1}(z⃗ + e⃗, t))}^{2}
(119) q_{n} = p_{n}(z⃗ + e⃗, t) − p_{n}(z⃗, t) − z̄_{1} · (p_{1}(z⃗ + e⃗, t) − p_{1}(z⃗, t))
−z̄_{1}Re(p_{2}(z⃗ + e⃗, t) − p_{2}(z⃗, t))
−2iz̄_{1}x_{3}Re(p_{3}(z⃗ + e⃗, t) − p_{3}(z⃗, t))
−iz̄_{1} · (Re(p_{3}(z⃗ + e⃗, t)))^{2}
−(z_{1} + x_{2} + it_{1} + ix_{3}^{2}) · \overline{(p_{1}(z⃗ + e⃗, t) − p_{1}(z⃗, t))}
−\overline{(p_{1}(z⃗ + e⃗, t))}p_{1}(z⃗ + e⃗, t)
−\overline{(p_{1}(z⃗ + e⃗, t))} · (Re(p_{2}(z⃗ + e⃗, t)) + iP_{1}(t))
−\overline{(p_{1}(z⃗ + e⃗, t))} · (2ix_{3}Re(p_{3}(z⃗ + e⃗, t)))
−\overline{(p_{1}(z⃗ + e⃗, t))} · (i(Re(p_{3}(z⃗ + e⃗, t)))^{2}).

To outline the role of q⃗ in the argument, the next step (Theorem 7.10) will suppose p⃗(z_{1}, …, z_{n}, w_{1}, …, w_{k}) is complex analytic on Δ_{ρ}, and |e⃗|_{κ} is small enough so that (z⃗, t) ∈ Δ_{κ} ⟹ (z⃗ + e⃗, t) ∈ Δ_{ρ}, and so q⃗ is a real analytic function for (z_{1}, z̄_{1}, x, t) ∈ D_{κ}. The quantity q⃗ can be thought of as an analogue of the RHS of Equation (23); it is the higher degree part of the new defining equations, but expressed in terms of the old coordinates. If q⃗(z_{1}, z̄_{1}, x, t) happens to be identically zero, then the manifold ; M has been brought to normal form by the functions p⃗. Otherwise, the degree of q⃗ is at least 2d − 3 by the construction of the solution p⃗, and defining q⃗(z_{1}, ζ, x, t) by (116–119), with ζ formally substituted for z̄_{1} and allowing complex x and t, the norm |q⃗|_{κ} can be bounded in terms of the norm of e⃗. Then later, in the Proof of Theorem 7.14, converting q⃗(z_{1}, z̄_{1}, x, t) into an expression in z̃_{1}, ¯ z̃_{1}, x̃, t̃, and equating it to the z̃ polynomial expression in (78) gives the defining equations of ; M in the (z̃, w̃) coordinate system, in analogy with the substitution in the next step after Equation (23).

The Ñ = N case of the following Lemma is proved in [C_{4}] (Lemma 4.1.).

Lemma 7.9. Let f = (f_{1}, …, fÑ) : 𝔻_{r} → ℂÑ be a holomorphic map with

max_{j=1,...,N} \{∑_{ℓ=1}^{Ñ} sup_{Z⃗∈𝔻_{r}} |\frac{df_{ℓ}}{dZ_{j}} (Z⃗)|\} ≤ K.

Then, for Z⃗ = (Z_{1}, …, Z_{N}), Z⃗′ = (Z′_{1}, …, Z′_{N}) ∈ 𝔻_{r},

∑_{ℓ=1}^{Ñ} |f_{ℓ}(Z⃗′) − f_{ℓ}(Z⃗)| ≤ K ∑_{j=1}^{N} |Z′_{j} − Z_{j}|.

Theorem 7.10. There are some constants c_{2} > 0 and δ_{1} > 0 (depending on m) such that if \frac{1}{2} < κ < r ≤ 1, and e⃗ is as in Theorem 7.6, with |e⃗|_{r} ≤ δ_{1}(r − κ), then

|q⃗|_{κ} ≤ \frac{c_{2}|e⃗|_{r}^{2}}{(r − κ)^{5}} .

Proof. Let ρ = \frac{1}{2}(r + κ). Note that if δ_{1} ≤ \frac{1}{2}, the formal series for q⃗ is convergent on D_{κ}, since for (z_{1}, ζ, x, t) ∈ D_{κ} ⊆ D_{r}, |x_{σ} + iE_{σ}| < κ + δ_{1}(r − κ) ≤ κ + (ρ − κ) = ρ, |(x_{2(τ−m+2)} + ix_{2(τ−m+2)+1})ζ + e_{τ}| < 2κ^{2}+(ρ−κ) < 2κ^{2}+(ρ−κ)(2(ρ+κ)) = 2ρ^{2}, and similarly |ζ^{2}+e_{n−1}| < ρ^{2} and |(z_{1} + x_{2} + it_{1} + ix_{3}^{2})ζ + e_{n}| < 3ρ^{2} + ρ^{3}, so (z⃗ + e⃗, t) ∈ Δ_{ρ}, which is contained in the domain of p⃗ by Theorem 7.6. The N = n + k, Ñ = 1, 𝔻_{r} = Δ_{ρ} case of Lemma 7.9 applies to p_{ℓ} : Δ_{ρ} → ℂ, with

max_{j=1,...,n, α=1,...,k} \{∥\frac{dp_{ℓ}}{dz_{j}} ∥_{ρ} , ∥\frac{dp_{ℓ}}{dw_{α}} ∥_{ρ}\} ≤ K = \frac{c_{1}|e⃗|_{r}}{(r − ρ)^{3}}

by Corollary 7.8, and Z⃗ = (z⃗, t), Z⃗′ = (z⃗ + e⃗, t) ∈ Δ_{ρ}, so the conclusion is:

|p_{ℓ}(z⃗ + e⃗, t) − p_{ℓ}(z⃗, t)| ≤ K(|E_{2}|_{r} + ⋯ + |e_{n}|_{r})
= \frac{c_{1}|e⃗|_{r}}{(r − ρ)^{3}} |e⃗|_{r} = \frac{8c_{1}|e⃗|_{r}^{2}}{(r − κ)^{3}} < \frac{2c_{1}|e⃗|_{r}^{2}}{(r − κ)^{5}} .

This provides bounds for the differences that appear in (116–119), and the remaining terms are the products, where we can also use \frac{1}{2} < κ < ρ < r ≤ 1 and the bound of Corollary 7.8 on the p_{ℓ} and P_{1} factors. For example, in a case of (117) where m > 4, part of the expression is the product:

sup_{D_{κ}} |(p_{1}(z⃗ + e⃗, t))′ \frac{p_{2(τ−m+2)}(z⃗ + e⃗, t) + (p_{2(τ−m+2)}(z⃗ + e⃗, t))′}{2} |

≤ ∥p_{1}∥_{ρ} · ∥p_{2(τ−m+2)}∥_{ρ} ≤ (\frac{c_{1}|e⃗|_{r}}{(r − ρ)^{2}} )^{2} = \frac{16(c_{1})^{2}|e⃗|_{r}^{2}}{(r − κ)^{4}} < \frac{8(c_{1})^{2}|e⃗|_{r}^{2}}{(r − κ)^{5}} .

Bounds for all of the other terms that are products of p_{ℓ} or P_{1} can be found similarly, except for the last line in (119), which is a product of three quantities:

sup_{D_{κ}} |(p_{1}(z⃗ + e⃗, t))′ · i (\frac{p_{3}(z⃗ + e⃗, t) + (p_{3}(z⃗ + e⃗, t))′}{2} )^{2}|

≤ ∥p_{1}∥_{ρ} · ∥p_{3}∥_{ρ}^{2} ≤ (\frac{c_{1}|e⃗|_{r}}{(r − ρ)^{2}} )^{3} = \frac{64(c_{1})^{3}|e⃗|_{r}^{3}}{(r − κ)^{6}}
≤ \frac{64(c_{1})^{3}|e⃗|_{r}^{2}δ_{1}(r − κ)}{(r − κ)^{6}} ≤ \frac{32(c_{1})^{3}|e⃗|_{r}^{2}}{(r − κ)^{5}} .

The cubic nonlinear term (from the iz̄_{1}x_{3}^{2} term in the normal form) in the last step of the above Proof is the only reason for the exponent 5 on \frac{1}{r−κ} in the statement of Theorem 7.10. There were no such cubic terms in the problems considered by [C_{4}] or [C_{6}], but the analysis will proceed in a similar way.

The following Lemma on inverse functions will be used twice, in the construction of the new coordinate system and the new defining equations; a proof by a standard iteration procedure is sketched in ([C_{4}]).

Lemma 7.11. Suppose 0 < R_{2,ℓ} < R_{1,ℓ} for ℓ = 1, …, N, so that

𝔻^{2} = 𝔻_{(R_{2,1},...,R_{2,N})} ⊆ 𝔻^{1} = 𝔻_{(R_{1,1},...,R_{1,N})}.

Let f(Z⃗) = (f_{1}(Z_{1}, …, Z_{N}), …, f_{N}(Z_{1}, …, Z_{N})) be holomorphic on 𝔻^{1}, with

max_{j=1,...,N} \{∑_{ℓ=1}^{N} sup_{Z⃗∈𝔻^{1}} |\frac{df_{ℓ}}{dZ_{j}} (Z⃗)|\} ≤ K < 1,

and

∑_{ℓ=1}^{N} sup_{Z⃗∈𝔻^{2}} |f_{ℓ}(Z⃗)| ≤ (1 − K) min_{ℓ=1,...,N}{R_{1,ℓ} − R_{2,ℓ}}.

Then, given W⃗ ∈ 𝔻^{2}, there exists a unique solution Z⃗ ∈ 𝔻^{1} of the equation

W⃗ = Z⃗ + f(Z⃗),

and this solution satisfies

∑_{ℓ=1}^{N} |Z_{ℓ} − W_{ℓ}| ≤ \frac{1}{1 − K} ∑_{ℓ=1}^{N} |f_{ℓ}(W⃗)|.

Theorem 7.12. There is some constant δ_{2} > 0 (depending on m) so that for any radius lengths \frac{1}{2} < κ < r ≤ 1, and e⃗, p⃗ as in Theorem 7.6, with |e⃗|_{r} ≤ δ_{2}(r − κ)^{3} and ρ = \frac{1}{2}(r + κ), the transformation

Ψ : (z⃗, w⃗) = (z_{1}, …, z_{n}, w_{1}, …, w_{k})^{T} ↦
(z̃, w̃) = (z_{1} + p_{1}(z⃗, w⃗), …, z_{n} + p_{n}(z⃗, w⃗), w_{1} + P_{1}(w⃗), w_{2}, …, w_{k})^{T}

has a holomorphic inverse ψ((z̃, w̃)) = (z⃗, w⃗) such that if (z̃, w̃) ∈ Δ_{κ}, then ψ((z̃, w̃)) ∈ Δ_{ρ}.

Proof. By Corollary 7.8,

max_{j=1,...,n} \{∥\frac{dP_{1}}{dz_{j}} ∥_{ρ} + ∑_{k=1}^{n} ∥\frac{dp_{k}}{dz_{j}} ∥_{ρ}\} ≤ \frac{c_{1}|e⃗|_{r}}{(r − ρ)^{3}}

< \frac{c_{1}δ_{2}(r − κ)^{3}}{(r − ρ)^{3}} = 8δ_{2}c_{1} ≤ \frac{1}{2} = K,

if δ_{2} ≤ \frac{1}{16c_{1}} , and similarly,

max_{α=1,...,k} \{∥\frac{dP_{1}}{dw_{α}} ∥_{ρ} + ∑_{ℓ=1}^{n} ∥\frac{dp_{ℓ}}{dw_{α}} ∥_{ρ}\} ≤ K.

Also by Corollary 7.8,

∥P_{1}∥_{κ}+∑_{ℓ=1}^{n} ∥p_{ℓ}∥_{κ} ≤ \frac{(n + 1)c_{1}|e⃗|_{r}}{(r − κ)^{2}} ≤ (n+1)c_{1}δ_{2}(r−κ) ≤ (1−K)(ρ−κ),

if δ_{2} ≤ \frac{1}{4(n+1)c_{1}} . The hypotheses of Lemma 7.11 are satisfied with Δ_{κ} ⊆ Δ_{ρ}, and R_{1,ℓ} − R_{2,ℓ} ≥ ρ − κ, so given (z̃, w̃) ∈ Δ_{κ}, there exists a unique (z⃗, w⃗) ∈ Δ_{ρ} such that (z̃, w̃) = (z_{1} + p_{1}(z⃗, w⃗), …, z_{n} + p_{n}(z⃗, w⃗), w_{1} + P_{1}(w⃗), w_{2}, …, w_{k})^{T}. This defines ψ so that Ψ ∘ ψ is the identity map on Δ_{κ}.

For (z_{1}, ζ, x, t) ∈ D_{R_{1}} ⊆ ℂ^{m+k}, define z^{c} ∈ ℂ^{n} by:

z^{c} = (z_{1}, x_{2} + iE_{2}(z_{1}, ζ, x, t), …, x_{m−1} + iE_{m−1}(z_{1}, ζ, x, t),
…, (x_{2(τ−m+2)} + ix_{2(τ−m+2)+1})ζ + e_{τ}(z_{1}, ζ, x, t), …,

ζ^{2} + e_{n−1}(z_{1}, ζ, x, t), (z_{1} + x_{2} + it_{1} + ix_{3}^{2})ζ + e_{n}(z_{1}, ζ, x, t))^{T},

and define a map Φ : D_{R_{1}} → ℂ^{m+k} by

Φ(z_{1}, ζ, x, t) = (Φ_{1}(z_{1}, ζ, x, t), …, Φ_{m+k}(z_{1}, ζ, x, t))
= (z_{1} + p_{1}(z^{c}, t), ζ + (p_{1}(z^{c}, t))′,

x_{2} + \frac{1}{2}(p_{2}(z^{c}, t) + (p_{2}(z^{c}, t))′), …,

x_{m−1} + \frac{1}{2}(p_{m−1}(z^{c}, t) + (p_{m−1}(z^{c}, t))′),

t_{1} + P_{1}(t), t_{2}, …, t_{k}).

Theorem 7.13. There is some constant δ_{3} > 0 (depending on m) so that for any radius lengths \frac{1}{2} < r′ < r ≤ 1, and e⃗, p⃗ as in Theorem 7.6, with |e⃗|_{r} ≤ δ_{3}(r − r′)^{3}, and κ = r′ + \frac{1}{3}(r − r′), the transformation Φ : (z_{1}, ζ, x, t) ↦ (z̃_{1}, ζ̃, x̃, t̃) has a holomorphic inverse ϕ(z̃_{1}, ζ̃, x̃, t̃) = (z_{1}, ζ, x, t) such that if (z̃_{1}, ζ̃, x̃, t̃) ∈ D_{r′}, then ϕ(z̃_{1}, ζ̃, x̃, t̃) ∈ D_{κ}.

Proof. Let ρ = r′+ \frac{2}{3}(r−r′), so κ−r′ = ρ−κ = r−ρ = \frac{1}{3}(r−r′) < \frac{1}{6}, and let r̄ = \frac{1}{2}(r + r′), so \frac{1}{2} < r′ < κ < r̄ < ρ < r ≤ 1. If (z_{1}, ζ, x, t) ∈ Dr̄, and δ_{3} ≤ \frac{2}{3}, then |E_{2}(z_{1}, ζ, x, t)| ≤ δ_{3}(r − r′)^{3} = 216δ_{3}(ρ − r̄)^{3} < \frac{216}{12^{2}}δ_{3}(ρ − r̄) ≤ ρ − r̄, and similarly |e_{n−1}(z_{1}, ζ, x, t)| < ρ^{2} − r̄^{2}, etc., so (z^{c}, t) ∈ Δ_{ρ}, and p⃗(z^{c}, t) and Φ are well-defined and holomorphic on Dr̄. Using Cauchy’s estimate as in Corollary 7.8, for (z_{1}, ζ, x, t) ∈ D_{κ},

|\frac{d}{dz_{1}} p_{2}(z^{c}, t)| ≤ \frac{|p_{2}(z^{c}, t)|r̄}{r̄ − κ} ≤ \frac{∥p_{2}∥_{ρ}}{\frac{1}{2}(ρ − κ)}
≤ \frac{2c_{1}|e⃗|_{r}}{(ρ − κ)(r − ρ)^{2}} = \frac{54c_{1}|e⃗|_{r}}{(r − r′)^{3}} .

Similarly, the derivative of each term, p_{1}(z^{c}, t), p_{σ}(z^{c}, t), (p_{1}(z^{c}, t))′, (p_{σ}(z^{c}), t)′, P_{1}(t), with respect to each variable z_{1}, ζ, x_{σ}, t_{α} is bounded by a comparable quantity, so there is some constant c_{3} > 0 (depending

102 UNFOLDING CR SINGULARITIES

on m) so that

max_{j=2,...,m−1, α=1,...,k}

\{|\frac{dp_{1}(z^{c}, t)}{dz_{1}} |_{κ}+|\frac{d((p_{1}(z^{c}, t))′)}{dz_{1}} |_{κ}+∑_{σ=2}^{m−1} |\frac{d(\frac{1}{2}(p_{σ}(z^{c}, t) + (p_{σ}(z^{c}, t))′))}{dz_{1}} |_{κ},

|\frac{dp_{1}(z^{c}, t)}{dζ}|_{κ}+|\frac{d((p_{1}(z^{c}, t))′)}{dζ}|_{κ}+∑_{σ=2}^{m−1} |\frac{d(\frac{1}{2}(p_{σ}(z^{c}, t) + (p_{σ}(z^{c}, t))′))}{dζ}|_{κ} ,

|\frac{dp_{1}(z^{c}, t)}{dx_{j}} |_{κ}+|\frac{d((p_{1}(z^{c}, t))′)}{dx_{j}} |_{κ}+∑_{σ=2}^{m−1} |\frac{d(\frac{1}{2}(p_{σ}(z^{c}, t) + (p_{σ}(z^{c}, t))′))}{dx_{j}} |_{κ} ,

|\frac{dp_{1}(z^{c}, t)}{dt_{α}} |_{κ}+|\frac{d((p_{1}(z^{c}, t))′)}{dt_{α}} |_{κ}+∑_{σ=2}^{m−1} |\frac{d(\frac{1}{2}(p_{σ}(z^{c}, t) + (p_{σ}(z^{c}, t))′))}{dt_{α}} |_{κ}

+ |\frac{dP_{1}(t)}{dt_{α}} |_{κ}\}

≤ \frac{c_{3}|e⃗|_{r}}{(r − r′)^{3}} ≤ c_{3}δ_{3} ≤ \frac{1}{2},

if δ_{3} ≤ \frac{1}{2c_{3}} . It also follows from Corollary 7.8 that |p_{1}(z^{c}, t)|_{r′} + |(p_{1}(z^{c}, t))′|_{r′}

+ (∑_{σ=2}^{m−1} |\frac{1}{2}(p_{σ}(z^{c}, t) + (p_{σ}(z^{c}, t))′)|_{r′}) + |P_{1}(t)|_{r′}

≤ 2∥p_{1}∥_{ρ} + (∑_{σ=2}^{m−1} ∥p_{σ}∥_{ρ}) + ∥P_{1}∥_{ρ}
≤ \frac{(m + 1)c_{1}|e⃗|_{r}}{(r − ρ)^{2}} ≤ \frac{(m + 1)c_{1}δ_{3}(r − r′)^{3}}{(r − ρ)^{2}}
= 9(m + 1)c_{1}δ_{3}(r − r′) ≤ \frac{1}{2}(κ − r′),

if δ_{3} ≤ \frac{1}{54(m+1)c_{1}} . So, by Lemma 7.11, given (z̃_{1}, ζ̃, x̃, t̃) ∈ D_{r′}, there exists a unique (z_{1}, ζ, x, t) ∈ D_{κ} such that (z̃_{1}, ζ̃, x̃, t̃) = Φ(z_{1}, ζ, x, t).

By inspection of the form of Φ, for (z_{1}, ζ, x, t) ∈ D_{κ},

Φ(z_{1}, ζ, x, t) = (z̃_{1}, ζ̃, x̃, t̃) ⟹ Φ(ζ̄, z̄_{1}, x̄, t̄) = (\overline{ζ̃}, ¯ z̃_{1}, ¯ x̃, \overline{t̃}). If, further, (z̃_{1}, ζ̃, x̃, t̃) = (\overline{ζ̃}, ¯ z̃_{1}, ¯ x̃, \overline{t̃}) ∈ D_{r′}, then (z_{1}, ζ, x, t) = (ζ̄, z̄_{1}, x̄, t̄) by uniqueness of the inverse. In particular, if |z̃_{1}| < r′, and for σ = 2, …, m − 1, x̃_{σ} is real and |x̃_{σ}| < r′, and for α = 1, …, k, t̃_{α} is real and |t̃| < r′, then ϕ(z̃_{1}, ¯ z̃_{1}, x̃, t̃) is of the form (z_{1}, z̄_{1}, x, t) for some z_{1} with |z_{1}| < κ, x real with |x_{σ}| < κ, and t real with |t_{α}| < κ. Such (z_{1}, x, t) is unique, given (z̃_{1}, x̃, t̃): suppose there were (z_{1}^{0}, x^{0}, t^{0}) with |z_{1}^{0}| < κ, |x_{σ}^{0}| < κ, x^{0} real, |t_{α}^{0}| < κ, t^{0} real, such that

z̃_{1} = Φ_{1}(z_{1}^{0}, \overline{z_{1}^{0}}, x^{0}, t^{0})
x̃_{σ} = Φ_{σ+1}(z_{1}^{0}, \overline{z_{1}^{0}}, x^{0}, t^{0}), σ = 2, …, m − 1,
t̃_{α} = Φ_{m+α}(z_{1}^{0}, \overline{z_{1}^{0}}, x^{0}, t^{0}), α = 1, …, k.

Then the second component Φ_{2}(z_{1}^{0}, \overline{z_{1}^{0}}, x^{0}, t^{0}) can be calculated to have some value ζ̃, so Φ(z_{1}^{0}, \overline{z_{1}^{0}}, x^{0}, t^{0}) = (z̃_{1}, ζ̃, x̃, t̃). By the formula for Φ, ζ̃ = ¯ z̃_{1}, so (z̃_{1}, ζ̃, x̃, t̃) ∈ D_{r′} and (z_{1}^{0}, \overline{z_{1}^{0}}, x^{0}, t^{0}) = ϕ(z̃_{1}, ζ̃, x̃, t̃) = ϕ(z̃_{1}, ¯ z̃_{1}, x̃, t̃) = (z_{1}, z̄_{1}, x, t), so we can conclude from the uniqueness of Lemma 7.11 that z_{1}^{0} = z_{1}, x^{0} = x, and t^{0} = t.

Theorem 7.14. There exist constants c_{4} > 0 and δ_{4} > 0 (depending on m) such that for any \frac{1}{2} < r′ < r ≤ 1 (with κ, ρ as in the previous Theorem), and any e⃗ as in Theorem 7.6 with |e⃗|_{r} ≤ δ_{4}(r − r′)^{3}, there exist a holomorphic map

Ψ : Δ_{ρ} → ℂ^{n+k} : (z_{1}, …, z_{n}, w_{1}, …, w_{k})^{T} ↦ (z̃_{1}, …, z̃_{n}, w̃_{1}, …, w̃_{k})^{T},

with a holomorphic inverse ψ : Δ_{κ} → Δ_{ρ}, and a holomorphic map ẽ = (Ẽ_{2}, …, ẽ_{n}) : D_{r′} → ℂ^{n−1}, such that the defining equations for ; M are

ỹ_{σ} = Im(z̃_{σ}) = Ẽ_{σ}(z̃_{1}, ¯ z̃_{1}, x̃, t̃)
z̃_{τ} = (x̃_{2(τ−m+2)} + ix̃_{2(τ−m+2)+1})¯z̃_{1} + ẽ_{τ}(z̃_{1}, ¯ z̃_{1}, x̃, t̃)
z̃_{n−1} = ¯ z̃_{1}^{2} + ẽ_{n−1}(z̃_{1}, ¯ z̃_{1}, x̃, t̃)
z̃_{n} = (z̃_{1} + x̃_{2} + it̃_{1} + ix̃_{3}^{2})¯z̃_{1} + ẽ_{n}(z̃_{1}, ¯ z̃_{1}, x̃, t̃)
s_{α} = Im(w̃_{α}) = 0,

for |z̃_{1}| < r′, |x̃_{σ}| < r′, |t̃_{α}| < r′. Further, the degree of ẽ is at least 2d − 3, and

|ẽ|_{r′} ≤ \frac{c_{4}|e⃗|_{r}^{2}}{(r − r′)^{5}} .

Proof. Initially, choose δ_{4} ≤ min\{\frac{8}{3}δ_{1}, \frac{8}{27}δ_{2}, δ_{3}\}, so that Theorems 7.10, 7.12, 7.13 apply, and define Ψ, ψ, q⃗, and ϕ in terms of the given e⃗ and the functions p⃗ constructed in Theorem 7.6. Define ẽ to be the composite of holomorphic maps q⃗∘ϕ : D_{r′} → ℂ^{n−1}, so that by Theorem 7.10,

|ẽ|_{r′} ≤ |q⃗|_{κ} ≤ \frac{c_{2}|e⃗|_{r}^{2}}{(r − κ)^{5}} = \frac{c_{2}|e⃗|_{r}^{2}}{(\frac{2}{3}(r − r′))^{5}} .

Since ϕ(z̃_{1}, ¯ z̃_{1}, x̃, t̃) has no constant terms, and q⃗ has degree ≥ 2d − 3 by construction, ẽ(z̃_{1}, ¯ z̃_{1}, x̃, t̃) also has degree at least 2d − 3.

Given z̃_{1}, x̃, t̃ = (t̃_{1}, …, t̃_{k}) such that |z̃_{1}| < r′, x̃ is real with |x̃_{σ}| < r′, and t̃ is real with |t̃_{α}| < r′, define quantities z̃_{2}, …, z̃_{n} by:

(120)  z̃_{σ} = x̃_{σ} + iẼ_{σ}(z̃_{1}, ¯ z̃_{1}, x̃, t̃)
z̃_{t} = (x̃_{2(t−m+2)} + ix̃_{2(t−m+2)+1})¯z̃_{1} + ẽ_{t}(z̃_{1}, ¯ z̃_{1}, x̃, t̃)
z̃_{n−1} = ¯ z̃_{1}^{2} + ẽ_{n−1}(z̃_{1}, ¯ z̃_{1}, x̃, t̃)
z̃_{n} = (z̃_{1} + x̃_{2} + it̃_{1} + ix̃_{3}^{2})¯z̃_{1} + ẽ_{n}(z̃_{1}, ¯ z̃_{1}, x̃, t̃),

and denote z̃ = (z̃_{1}, z̃_{2}, …, z̃_{n})^{T}. The claim of the Theorem is that

ψ((z̃, t̃)) ∈ ; M.

If (δ_{4})^{2} ≤ \frac{32}{729c_{2}} , then

|ẽ|_{r′} ≤ \frac{c_{2}(δ_{4}(r − r′)^{3})^{2}}{(r − κ)^{5}} = c_{2}(δ_{4})^{2} \frac{3^{6}}{2^{5}} (κ − r′) ≤ κ − r′,

so (z̃, t̃) ∈ Δ_{κ}, the domain of ψ.

By Theorem 7.13, there exists a unique (z_{1}, x, t) (the first and last components of (z_{1}, z̄_{1}, x, t) = ϕ(z̃_{1}, ¯ z̃_{1}, x̃, t̃)) such that |z_{1}| < κ, x is real with |x_{σ}| < κ, t is real with |t_{α}| < κ, and

z̃_{1} = z_{1} + p_{1}(z_{1}, x_{2} + iE_{2}(z_{1}, z̄_{1}, x, t), …,
(z_{1} + x_{2} + it_{1} + ix_{3}^{2})z̄_{1} + e_{n}(z_{1}, z̄_{1}, x, t), t)
x̃_{σ} = x_{σ} + Re(p_{σ}(z_{1}, x_{2} + iE_{2}(z_{1}, z̄_{1}, x, t), …,
(z_{1} + x_{2} + it_{1} + ix_{3}^{2})z̄_{1} + e_{n}(z_{1}, z̄_{1}, x, t), t))
t̃_{1} = t_{1} + P_{1}(t)
t̃_{j} = t_{j}, j = 2, …, k.

Then, define quantities z_{2}, …, z_{n} by:

z_{σ} = x_{σ} + iE_{σ}(z_{1}, z̄_{1}, x, t)
z_{τ} = (x_{2(τ−m+2)} + ix_{2(τ−m+2)+1})z̄_{1} + e_{τ}(z_{1}, z̄_{1}, x, t)
z_{n−1} = z̄_{1}^{2} + e_{n−1}(z_{1}, z̄_{1}, x, t)
z_{n} = (z_{1} + x_{2} + it_{1} + ix_{3}^{2})z̄_{1} + e_{n}(z_{1}, z̄_{1}, x, t),

and denote, as in (88) and (114), z⃗ = (z_{1}, x, …, (z_{1} +x_{2} +it_{1} +ix_{3}^{2})z̄_{1})^{T} and z⃗ + e⃗ = (z_{1}, z_{2}, …, z_{n})^{T} as in (79) and (115). Since |z_{1}| < κ < r,

|x_{σ}| < κ < r, and |t_{α}| < κ < r, (z⃗ + e⃗, t) ∈ ; M, and if δ_{4} ≤ \frac{4}{3}, then

|e⃗|_{κ} ≤ |e⃗|_{r} ≤ δ_{4}(r − r′)^{3} = δ_{4} · 27(ρ − κ)^{3} < δ_{4} \frac{27}{6^{2}}(ρ − κ) ≤ (ρ − κ),

so (z⃗ + e⃗, t) ∈ Δ_{ρ}, which is contained in the domain of p⃗.

Ψ(z⃗ + e⃗, t)

= (z_{1} + p_{1}((z⃗ + e⃗, t)), …, z_{n} + p_{n}((z⃗ + e⃗, t)), t_{1} + P_{1}(t), t_{2}, …, t_{k})^{T}
= (z̃_{1}, …, x̃_{σ} + iE_{σ}(z_{1}, z̄_{1}, x, t) + iIm(p_{σ}((z⃗ + e⃗, t))), …,

\overline{(z̃_{1} − p_{1}((z⃗ + e⃗, t)))} · (x̃_{2(τ−m+2)} − Re(p_{2(τ−m+2)}((z⃗ + e⃗, t))))

+i\overline{(z̃_{1} − p_{1}((z⃗ + e⃗, t)))}·(x̃_{2(τ−m+2)+1} − Re(p_{2(τ−m+2)+1}((z⃗ + e⃗, t)))) +e_{τ}(z_{1}, z̄_{1}, x, t) + p_{τ}((z⃗ + e⃗, t)), …

\overline{(z̃_{1} − p_{1}((z⃗ + e⃗, t)))}^{2} + e_{n−1}(z_{1}, z̄_{1}, x, t) + p_{n−1}((z⃗ + e⃗, t)),

\overline{(z̃_{1} − p_{1}((z⃗ + e⃗, t)))}·(z̃_{1} − p_{1}((z⃗ + e⃗, t)) + x̃_{2} − Re(p_{2}((z⃗ + e⃗, t))))

+i\overline{(z̃_{1} − p_{1}((z⃗ + e⃗, t)))} · (t̃_{1} − P_{1}(t))

+i\overline{(z̃_{1} − p_{1}((z⃗ + e⃗, t)))} · (x̃_{3} − Re(p_{3}((z⃗ + e⃗, t))))^{2} +e_{n}(z_{1}, z̄_{1}, x, t) + p_{n}((z⃗ + e⃗, t)), t̃)^{T}

= (z̃_{1}, …, x̃_{σ} + iQ_{σ}(z_{1}, z̄_{1}, x, t), …,

¯z̃_{1} · (x̃_{2(τ−m+2)} + ix̃_{2(τ−m+2)+1}) + q_{τ}(z_{1}, z̄_{1}, x, t), …, ¯z̃_{1}^{2} + q_{n−1}(z_{1}, z̄_{1}, x, t), ¯z̃_{1} · (z̃_{1} + x̃_{2} + it̃_{1} + ix̃_{3}^{2}) + q_{n}(z_{1}, z̄_{1}, x, t), t̃)^{T}

= (z̃_{1}, …, x̃_{σ} + iQ_{σ}(ϕ(z̃_{1}, ¯ z̃_{1}, x̃, t̃)), …,

¯z̃_{1} · (x̃_{2(τ−m+2)} + ix̃_{2(τ−m+2)+1}) + q_{τ}(ϕ(z̃_{1}, ¯ z̃_{1}, x̃, t̃)), …, ¯z̃_{1}^{2} + q_{n−1}(ϕ(z̃_{1}, ¯ z̃_{1}, x̃, t̃)), ¯z̃_{1} · (z̃_{1} + x̃_{2} + it̃_{1} + ix̃_{3}^{2}) + q_{n}(ϕ(z̃_{1}, ¯ z̃_{1}, x̃, t̃)), t̃)^{T}

= (z̃_{1}, …, x̃_{σ} + iẼ_{σ}(z̃_{1}, ¯ z̃_{1}, x̃, t̃), …,

¯z̃_{1} · (x̃_{2(τ−m+2)} + ix̃_{2(τ−m+2)+1}) + ẽ_{τ}(z̃_{1}, ¯ z̃_{1}, x̃, t̃), …, ¯z̃_{1}^{2} + ẽ_{n−1}(z̃_{1}, ¯ z̃_{1}, x̃, t̃), ¯z̃_{1} · (z̃_{1} + x̃_{2} + it̃_{1} + ix̃_{3}^{2}) + ẽ_{n}(z̃_{1}, ¯ z̃_{1}, x̃, t̃), t̃)^{T} = (z̃, t̃)

by construction of q⃗, ẽ, and z̃, and using the fact that p⃗ is a solution of (84–88). By the uniqueness of Theorem 7.12, ψ((z̃, t̃)) = (z⃗ + e⃗, t) ∈

; M.

7.4. Composition of approximate solutions. The previous Theorem’s quadratic estimate on the size of ẽ in terms

of e⃗ allows for the rapid convergence of a sequence of approximations. A couple technical Lemmas will be needed to measure the behavior of composite mappings. Theorem 7.21, which is the last step in proving Main Theorem 6.5, uses these Lemmas and the estimates of the previous Subsection to prove convergence of a sequence of transformations, following the ideas of [Moser].

Notation 7.15. For R_{1} > 0 and a (n + k) × (n + k) matrix of complex valued functions F = (F_{ℓj}((z⃗, w⃗))) on Δ_{R_{1}}, define

|||F |||_{R_{1}} = max_{j=1,...,n+k} {∑_{ℓ=1}^{n+k} sup_{(z⃗,w⃗)∈Δ_{R_{1}}} |F_{ℓj}((z⃗, w⃗))|} .

This “maximum column sum” norm appeared already, in Corollary 7.8 and Lemmas 7.9, 7.11, in the case where F = Df = D_{(z⃗,w⃗)}f, the Jacobian matrix of some map f : Δ_{R_{1}} → ℂ^{n+k} at (z⃗, w⃗) ∈ Δ_{R_{1}}.

The 3 × 3 case of the following Lemma was proved in [C_{4}].

Lemma 7.16. If |||A|||_{R_{1}} < 1, then 𝟙 + A is invertible (where 𝟙 is the (n + k) × (n + k) identity matrix), and

|||(𝟙 + A)^{−1}|||_{R_{1}} ≤ \frac{1}{1 − |||A|||_{R_{1}}} .

Also, the following elementary fact from the calculus of one real variable will be used.

Lemma 7.17. If μ_{ℓ} is a sequence such that 0 ≤ μ_{ℓ} < 1 and ∑_{ℓ=0}^{∞} μ_{ℓ}

is a convergent series, then the sequence of partial products

∏_{ℓ=0}^{N} \frac{1}{1 − μ_{ℓ}}

is bounded above by some positive limit.

Notation 7.18. Define a sequence \{1, \frac{3}{4}, \frac{4}{6}, \frac{5}{8}, …\} indexed by ν = 0, 1, 2, 3, …, by the formula

r_{ν} = \frac{1}{2} (1 + \frac{1}{ν + 1}) .

Note that \frac{1}{2} < r_{ν} ≤ 1, and the sequence is decreasing, with

r_{ν} − r_{ν+1} = \frac{1}{2(ν + 1)(ν + 2)} ≤ \frac{1}{4},
\frac{r_{ν+1} − r_{ν+2}}{r_{ν} − r_{ν+1}} = \frac{ν + 1}{ν + 3} ≥ \frac{1}{3}.

Notation 7.19. Define κ_{ν} = r_{ν+1}+ \frac{1}{3}(r_{ν} −r_{ν+1}), ρ_{ν} = r_{ν+1}+ \frac{2}{3}(r_{ν} − r_{ν+1}), as in Theorem 7.13.

Recall that given η_{0} > 0, there is some scaling transformation (113) so that ; M ∩ Δ_{1} is defined by (75), with e⃗ holomorphic on D_{1}, degree d ≥ 4 (with the cubic terms of ; M already in normal form as in Remark 7.7), and |e⃗|_{1} ≤ η_{0}.

Notation 7.20. Denote e⃗_{0} = e⃗ (so |e⃗_{0}|_{r_{0}} = |e⃗|_{1} ≤ η_{0}), and inductively define the formal series e⃗_{ν+1}(z_{1}, ζ, x, t) in terms of e⃗_{ν}(z_{1}, ζ, x, t), by the e⃗ ↦ ẽ procedure of Theorem 7.14, with r = r_{ν}, r′ = r_{ν+1}. Each e⃗_{ν} defines, as in the previous Theorems, functions p⃗_{ν}, q⃗_{ν}, Ψ_{ν}, ψ_{ν}, ϕ_{ν}, and the degree of e⃗_{ν} is denoted d_{ν}.

Also recall that the degree d_{ν+1} of e⃗_{ν+1} is at least 2d_{ν} − 3; it can be checked that this, together with d_{0} = d ≥ 4, implies d_{ν} ≥ 2^{ν} + 3.

The plan is to show that the bound for e⃗_{ν} in the hypothesis of Theorem 7.14 holds for all ν, to get a sequence of transformations ψ_{ν} : Δ_{κ_{ν}} → Δ_{ρ_{ν}}, so that the composition ψ_{0}∘…∘ψ_{ν−1}∘ψ_{ν} : Δ_{κ_{ν}} → Δ_{ρ_{0}} is well-defined, e⃗_{ν} is holomorphic on D_{r_{ν}}, and lim_{ν→∞} |e⃗_{ν}|_{r_{ν}} = 0.

Theorem 7.21. There exists η_{0} > 0 (depending on m) so that if e⃗_{0} and ; M are as described above, then there exists a holomorphic transformation ψ : Δ_{\frac{1}{2}} → ℂ^{n+k}, with a holomorphic inverse Ψ, and such that if (z̃, t̃) ∈ ; M ∩ Δ_{\frac{1}{2}} , then ψ((z̃, t̃)) ∈ ; M.

Proof. Let δ_{5} = min\{16δ_{4}, \frac{1}{243c_{4}} \}, and choose

0 < η_{0} < min \{\frac{δ_{5}}{1024}, \frac{1}{1728c_{1}} \} .

It will be shown that

|e⃗_{ν}|_{r_{ν}} ≤ δ_{5}(r_{ν} − r_{ν+1})^{5} ⟹ |e⃗_{ν+1}|_{r_{ν+1}} ≤ δ_{5}(r_{ν+1} − r_{ν+2})^{5}.

By Theorem 7.14, |e⃗_{ν}|_{r_{ν}} ≤ δ_{5}(r_{ν} − r_{ν+1})^{5} ≤ δ_{4}(r_{ν} − r_{ν+1})^{3} and |e⃗_{ν}|_{r_{ν}} ≤ \frac{1}{243c_{4}} (r_{ν} − r_{ν+1})^{5} imply

|e⃗_{ν+1}|_{r_{ν+1}} ≤ \frac{c_{4}|e⃗_{ν}|_{r_{ν}}^{2}}{(r_{ν} − r_{ν+1})^{5}} ≤ \frac{1}{243}|e⃗_{ν}|_{r_{ν}};

this already suggests a geometric decrease in the sequence of norms. Then, using the properties of the sequence r_{ν},

\frac{1}{243}|e⃗_{ν}|_{r_{ν}} ≤ \frac{1}{243}δ_{5}(r_{ν} − r_{ν+1})^{5} ≤ δ_{5}(r_{ν+1} − r_{ν+2})^{5},

which proves the claimed implication. Using this as an inductive step, and starting the induction with |e⃗_{0}|_{r_{0}} ≤ η_{0} < \frac{1}{1024}δ_{5} = δ_{5}(r_{0} − r_{1})^{5}, the hypothesis of Theorem 7.14 is satisfied for all ν. The first of three conclusions from Theorem 7.14 is that e⃗_{ν} is holomorphic on D_{r_{ν}}, with degree d_{ν} ≥ 2^{ν} + 3, and |e⃗_{ν}|_{r_{ν}} ≤ 243^{−ν}η_{0}. Secondly, ψ_{0} ∘ … ∘ ψ_{ν} is a well-defined holomorphic map Δ_{κ_{ν}} → Δ_{ρ_{0}}, and Ψ_{ν} ∘ … ∘ Ψ_{0} is welldefined and holomorphic on the image (ψ_{0} ∘ … ∘ ψ_{ν})(Δ_{κ_{ν}}), so that Ψ_{ν} ∘ … ∘ Ψ_{0} ∘ ψ_{0} ∘ … ∘ ψ_{ν} is the identity on Δ_{κ_{ν}}. The third conclusion is that if |z̃_{1}| < r_{ν+1}, |x̃_{σ}| < r_{ν+1}, and |t̃_{α}| < r_{ν+1}, and (z̃, t̃) is defined as in (120) with ẽ = e⃗_{ν+1}, then (ψ_{0} ∘ … ∘ ψ_{ν})((z̃, t̃)) ∈ ; M. For any (z⃗, w⃗) = (z_{1}, …, z_{n}, w_{1}, …, w_{k}) ∈ Δ_{\frac{1}{2}} , the sequence (depending on ν) (ψ_{0} ∘…∘ψ_{ν−1} ∘ψ_{ν})((z⃗, w⃗)) is contained in Δ_{ρ_{0}} = Δ_{11/12}. The following argument, beginning with several applications of Lemma 7.9, shows this sequence is a Cauchy sequence, and converges to some value ψ((z⃗, w⃗)).

∑_{ℓ=1}^{n+k} |(ψ_{0} ∘ … ∘ ψ_{ν+1})_{ℓ}((z⃗, w⃗)) − (ψ_{0} ∘ … ∘ ψ_{ν})_{ℓ}((z⃗, w⃗))|

= ∑_{ℓ=1}^{n+k} |(ψ_{0})_{ℓ}((ψ_{1} ∘ … ∘ ψ_{ν+1})((z⃗, w⃗)))
−(ψ_{0})_{ℓ}((ψ_{1} ∘ … ∘ ψ_{ν})((z⃗, w⃗)))|
≤ |||Dψ_{0}|||_{ρ_{1}} · ∑_{j=1}^{n+k} |(ψ_{1} ∘ … ∘ ψ_{ν+1})_{j}((z⃗, w⃗))
−(ψ_{1} ∘ … ∘ ψ_{ν})_{j}((z⃗, w⃗))|
(121) ≤ (∏_{ℓ=0}^{ν} |||Dψ_{ℓ}|||_{ρ_{ℓ+1}}) · ∑_{j=1}^{n+k} |(ψ_{ν+1})_{j}((z⃗, w⃗)) − (z⃗, w⃗)_{j}|,

where (ψ_{0})_{ℓ} denotes the ℓ^{th} output component of the vector valued function ψ_{0}, etc., and (z⃗, w⃗)_{j} = z_{j} for j = 1, …, n, or w_{j−n} for j = n + 1, …, n + k. By the estimate from Lemma 7.11, with f = p⃗_{ν+1} and K = \frac{1}{2} from the Proof of Theorem 7.12, and then using the bound for p⃗ from Corollary 7.8,

∑_{j=1}^{n+k} |(ψ_{ν+1})_{j}((z⃗, w⃗)) − (z⃗, w⃗)_{j}| ≤ \frac{1}{1 − \frac{1}{2}} ∑_{j=1}^{n+k} |(p⃗_{ν+1})_{j}((z⃗, w⃗))|

≤ 2 ∑_{j=1}^{n+k} ∥(p⃗_{ν+1})_{j}∥_{\frac{1}{2}} ≤ 2 ∑_{j=1}^{n+1} ∥(p⃗_{ν+1})_{j}∥_{ρ_{ν+1}} ≤ 2(n + 1)\frac{c_{1}|e⃗_{ν+1}|_{r_{ν+1}}}{(r_{ν+1} − ρ_{ν+1})^{2}}

= 18(n + 1)\frac{c_{1}|e⃗_{ν+1}|_{r_{ν+1}}}{(r_{ν+1} − r_{ν+2})^{2}} = 72(n + 1)c_{1}(ν + 2)^{2}(ν + 3)^{2}|e⃗_{ν+1}|_{r_{ν+1}}
≤ \frac{72(n + 1)c_{1}(ν + 2)^{2}(ν + 3)^{2}η_{0}}{243^{ν+1}} .

It follows from D_{(z⃗,w⃗)}ψ_{ℓ} = (𝟙+D_{ψ_{ℓ}((z⃗,w⃗))}p⃗_{ℓ})^{−1} and Lemma 7.16 that:

|||Dψ_{ℓ}|||_{ρ_{ℓ+1}} = |||(𝟙 + D_{ψ_{ℓ}((z⃗,w⃗))}p⃗_{ℓ})^{−1}|||_{ρ_{ℓ+1}}
≤ |||(𝟙 + Dp⃗_{ℓ})^{−1}|||_{ρ_{ℓ}} ≤ \frac{1}{1 − |||Dp⃗_{ℓ}|||_{ρ_{ℓ}}} .

Then, by Lemma 7.17, the product from (121) is bounded above by some constant c_{5} > 0, since by Corollary 7.8,

|||Dp⃗_{ℓ}|||_{ρ_{ℓ}} ≤ \frac{c_{1}|e⃗_{ℓ}|_{r_{ℓ}}}{(r_{ℓ} − ρ_{ℓ})^{3}} = \frac{27c_{1}|e⃗_{ℓ}|_{r_{ℓ}}}{(r_{ℓ} − r_{ℓ+1})^{3}} = 216(ℓ + 1)^{3}(ℓ + 2)^{3}c_{1}|e⃗_{ℓ}|_{r_{ℓ}}
≤ \frac{216(ℓ + 1)^{3}(ℓ + 2)^{3}c_{1}η_{0}}{243^{ℓ}} < 1,

and by the comparison:

∑_{ℓ=0}^{∞} |||Dp⃗_{ℓ}|||_{ρ_{ℓ}} ≤ ∑_{ℓ=0}^{∞} \frac{216(ℓ + 1)^{3}(ℓ + 2)^{3}c_{1}η_{0}}{243^{ℓ}} ,

the infinite series is convergent, with terms < 1.

The inequality

∑_{ℓ=1}^{n+k} |(ψ_{0} ∘ … ∘ ψ_{ν+1})_{ℓ}((z⃗, w⃗)) − (ψ_{0} ∘ … ∘ ψ_{ν})_{ℓ}((z⃗, w⃗))|

≤ \frac{72(n + 1)c_{1}c_{5}(ν + 2)^{2}(ν + 3)^{2}η_{0}}{243^{ν+1}}

is enough to show that the sequence of composite functions converges pointwise and uniformly to a function ψ on Δ_{\frac{1}{2}} .

8. Some other directions

1. The lists of normal forms from Section 5.1 stopped at Q ≡ 0, although some results on surfaces with a higher order contact with the complex tangent line appear in [H_{1}], [H_{2}], [Bharali]. The list of normal forms from Section 6.1 is also incomplete.

2. The calculations of normal forms of unfoldings of surfaces from Section 5.2 stopped at low degree, leaving O(4) or O(5) quantities not normalized. Presumably some higher degree terms in the normal forms could be analyzed by further calculations, but the term-by-term approach of that Section does become complicated.

3. In Example 5.13, the unfolding normal form (36) for generic elliptic points was observed to be similar to the flattened algebraic normal form result of [MW] for n-manifolds in ℂ^{n} under biholomorphic transformations. This leaves open the question: can the [MW] normal form (37) be achieved by a transformation in the subgroup U_{2,2,k} (27– 28), or if not, how is the unfolding classification different?

4. The local geometry of a manifold near a point in N_{j} (as in (2)), for m ≤ n, j > 1, where the tangent space contains a complex plane, has not been studied as much as the j = 1 case. Since the codimension of N_{2} is generally 4(n − m + 2), it could have a nonnegative expected dimension for m ≥ 8 and some n. It would be interesting to understand the local geometry of M near a point in N_{2}, and how such a manifold might be deformed so that other types of singularities in N_{1} or N_{2} appear.

5. The case \frac{2}{3}(n + 1) < m ≤ n, where M has high codimension, but the locus N_{1} has a positive expected dimension, is also interesting. Globally, there are characteristic class formulas ([Webster_{2}], [Webster_{3}], [C_{1}]) describing the non-isolated CR singularities, generalizing (3). The formulas of [Webster_{2}] involving parabolic singularities in the m = n case have not been generalized to other types of degenerate singularities. Locally, the normal forms in the nondegenerate case for \frac{2}{3}(n + 1) < m < n are known ([C_{6}]), but the normal forms for degenerate CR singularities are not, and may not be as simple as those in Proposition 6.2. Normal forms for unfolding could also become complicated.

6. Another less-explored case of CR singular m-submanifolds of ℂ^{n} is where m > n, so the generic point x of a submanifold in general position satisfies j = dim T_{x} ∩JT_{x} = m−n, and the CR singular points are where this dimension jumps. The expected codimension formula codim N_{j} = 2j(n − m + j) still holds for j ≥ m − n and m > n, and some j > m − n cases have been considered by [G].

7. The use of all real variables to parametrize the deformation seemed to be a more natural, or at least simpler, choice than using complex parameters, since the coefficients appearing in normal forms (for example, the Bishop invariant) can be real quantities, and a real time variable in the k = 1 case was useful in the visualization. A more general unfolding construction could use both complex and real deformation parameters. Then, for example, even for M with only N_{1} \ N_{2} singularities, the manifold ; M could be tangent to a complex plane.

Acknowledgments. The author’s research at IPFW was supported in part by a sabbatical semester. The author also thanks X. Gong (University of Wisconsin - Madison) for helpful comments on drafts of this paper.

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