FORMAL STABILITY OF THE CR CROSS-CAP

Adam Coffman

For m < n, any real analytic m-submanifold of complex n-space with a non-degenerate CR singularity is shown to be locally equivalent, under a formal holomorphic coordinate change, to a fixed real algebraic variety defined by linear and quadratic polynomials. The situation is analogous to Whitney’s stability theorem for cross-cap singularities of smooth maps. The formal coordinate change is obtained from a sequence of approximations whose norms are controlled only on polydiscs shrinking to a point, so the analytic classification remains an open question.

1. Introduction

In [Wh_{43}], H. Whitney described the parametrization (u, v) ↦ (u^{2}, v, uv) of the “cross-cap” as an example of a differentiable map from ℝ^{2} to ℝ^{3} with a singular point at the origin. He further demonstrated that for any sufficiently generic singularity of a map f from ℝ^{2} to ℝ^{3}, there exists a local change of coordinates with respect to which f has exactly that normal form: f(u, v) = (u^{2}, v, uv). The coordinate change is C^{r}, smooth, or real analytic in a neighborhood, as f is C^{4r+8}, smooth, or real analytic. This “stability theorem” was later generalized ([H]) to generic singularities of maps f : ℝ^{m} → ℝ^{n} for m ≤ n, with an analogous normal form. A crucial step in Whitney’s argument was the use of his famous lemma that a smooth, even function F (x) = F (−x) of one variable can be written as a smooth function of x^{2}: F (x) = g(x^{2}).

The objects of study in this paper are real m-submanifolds M of complex n-space ℂ^{n}. The occurence of a complex line in a tangent space T_{x}M when m ≤ n is called a “CR singularity,” or “complex tangent,” and some of the local and global geometric properties of these objects have analogues in the singularity theory of maps; the m = n case has been studied since [Bishop]. Here the m < n case is examined and compared with Whitney’s normal form theorem.

Specifically, the local geometry of M near the complex tangent is analyzed by establishing some non-degeneracy conditions and arriving at a normal form for the quadratic part of the defining equations. The main result is the following Proposition:

Proposition 1.1. Given m and n such that \frac{2}{3}(n + 1) ≤ m < n, there exists a real algebraic variety M̃^{m} in ℂ^{n} defined by quadratic and linear polynomials such that a germ of any real analytic m-submanifold M^{m} of ℂ^{n} at a suitably non-degenerate complex tangent is equivalent to M̃^{m} under a formal holomorphic coordinate change of ℂ^{n}.

The simplest case of the Proposition is m = 4, n = 5, which follows as a consequence of Theorem 3.16. The generalization to higher dimensions is outlined in Subsection 2.4. The result contrasts with the m = n case, where there can be higher order invariants of real analytic embeddings ([MW]). For 4-manifolds in ℂ^{5}, the corresponding variety M̃^{4} is defined by these equations in the variables z_{1} = x_{1} + iy_{1}, …, z_{5} = x_{5} + iy_{5}:

M̃^{4} = {z⃗ : y_{2} = y_{3} = 0, z_{4} = (z̄_{1} + x_{2} + ix_{3})^{2}, z_{5} = z_{1}(z̄_{1} + x_{2} + ix_{3})}.

Note there are no continuous invariants (again, unlike the m = n case, where the normal form depends on the value of Bishop’s invariant β ≥ 0), and the two quantities L̄^{2} and z_{1}L̄, where L̄ = z̄_{1} + x_{2} + ix_{3}, cannot be simultaneously transformed into monomials by a holomorphic coordinate change (unlike Whitney’s normal form).

A strict analogy with Whitney’s result would suggest that a change of coordinates can be found which converges in a neighborhood of 0, but there are holomorphic normal form problems where the formal and analytic classifications are different — [V], for example. Here, the existence of formal power series defining a coordinate change bringing the singularity to normal form is established by an iterative procedure, where the main step is to solve a linearized equation. If the defining functions are quadratic plus degree d and higher terms, a coordinate change defined using the solution of the linear equation eliminates terms up to about twice the degree. However, control over a certain norm of this solution will only be established on a polydisc significantly smaller than that on which the defining equations were given. Iterating the linearization makes this polydisc arbitrarily small, and no conclusion can be drawn regarding the convergence of the formal solution. Contributing to this shrinking is a rearrangement phenomenon specific to the calculus of several variables. As in Whitney’s calculations, some real analytic functions are split into even and odd parts, but the Proof of Theorem 3.16 will first rearrange series in z_{1}, z̄_{1}, x_{2}, x_{3} into series in z_{1}, L̄, x_{2}, x_{3}, and then find the (even + odd) decomposition in the L̄ variable. These even and odd parts may be real analytic on a smaller domain.

The effect of such a rearrangement on the polyradius of convergence of a power series will be analyzed in Section 3, to the extent that the details relate to the linearized normal form problem. Since the 4-manifold is not a uniqueness set for holomorphic functions in ℂ^{5}, there are divergent solutions of the linear equation, but, after a choice of normalization, a unique solution is constructed which converges on a small polydisc.

This paper is based on the main result of the author’s 1997 dissertation, [C], supervised by S. Webster.

Note: after this paper was accepted for publication, the author learned that the m = 4, n = 5 case of Proposition 1.1 has also been considered by [Beloshapka].

2. Coordinate changes, and normal forms for complex tangents

Attention will be focused on the m = 4, n = 5 case of a real m-submanifold of a complex n-manifold. These are the lowest dimensions m < n where complex tangents are “topologically stable”; for example, if a real surface in ℂ^{3} has a complex tangent plane at some point, this property is not shared with most nearby surfaces. When m < n, the real m-planes T such that dim_{ℂ} T ∩ iT ≥ j form a subvariety D_{j} of real codimension 2j(n − m + j) inside the grassmannian of real m-planes in ℂ^{n} ≅ ℝ^{2n}. The occurence of complex tangents of an immersion corresponds to the intersection of the Gauss map with D_{j}, and the immersion could be called “generic” if the immersed manifold’s Gauss map meets each stratum D_{j} \D_{j+1} transversely. So, generic immersions of M in ℂ^{n} are “totally real” outside a codimension 2(n − m + 1) subset of M, and if m < \frac{2}{3}(n + 1), then M is generically totally real everywhere. (This resembles the bounds in Whitney’s embedding and immersion theorems; see [Wh_{44}].) The case addressed by this paper is \frac{2}{3}(n + 1) ≤ m < n, and j = 1; only points where the tangent space contains exactly one complex line will be considered. In the m = 4, n = 5 case, the generic singularity is isolated. This Section establishes non-degeneracy conditions for complex tangents and proposes quadratic normal forms in the above dimension range. The initial assumption is that M = M^{4} is a 4-dimensional submanifold containing the origin of ℂ^{5}, defined as the zero set of real analytic functions in a neighborhood of the origin. 2.1. Holomorphic coordinate changes. It can be assumed that the tangent plane at the origin in ℂ^{5}, T = T0M, has four real dimensions and contains the complex line with coordinate z_{1} = x_{1} + iy_{1}. The remaining coordinates of ℂ^{5} can be chosen as z_{2}, …, z_{5}, so that T is defined by y_{2} = y_{3} = z_{4} = z_{5} = 0 and has coordinates z_{1}, x_{2}, and x_{3}. The complex linear transformations of ℂ^{5} that preserve T, written as 5×5 matrices acting on column vectors, are of the form

A = ( \begin{matrix}a_{1}^{1} & a_{1}^{2} & a_{1}^{3} & a_{1}^{4} & a_{1}^{5} \\ 0 & r_{2}^{2} & r_{2}^{3} & a_{2}^{4} & a_{2}^{5} \\ 0 & r_{3}^{2} & r_{3}^{3} & a_{3}^{4} & a_{3}^{5} \\ 0 & 0 & 0 & a_{4}^{4} & a_{4}^{5} \\ 0 & 0 & 0 & a_{5}^{4} & a_{5}^{5} \end{matrix} ) ,

where the a entries are complex, and the entries of the block R = ( \begin{matrix}r_{2}^{2} & r_{2}^{3} \\ r_{3}^{2} & r_{3}^{3} \end{matrix} )

are real.

Since these entries will be used in a coordinate change transforming the quadratic and cubic parts of the defining equations for M into a normal form, det(A) ≠ 0. If T is given some orientation, then the transformations preserving that orientation also satisfy det(R) > 0. For example, the map z_{3} ↦ −z_{3} is complex linear, but reverses the orientation of the real subspace.

Once the real submanifold M is placed with its tangent space at the origin as described, the geometry of interest is its interaction with the ambient complex analytic structure. To find features which are invariant under biholomorphic coordinate changes of ℂ^{5}, it suffices to consider local biholomorphisms with linear part preserving T. The transformation from z_{s} to z̃_{r} coordinates, r, s = 1, …, 5, and its holomorphic inverse, will be written as multivariable power series, using the summation convention:

(1)  z̃_{r} = a_{r}^{s}z_{s} + p_{r}^{st}z_{s}z_{t} + p_{r}^{stu}z_{s}z_{t}z_{u} + …,
(2)  z_{r} = b_{r}^{s}z̃_{s} + q_{r}^{st}z̃_{s}z̃_{t} + q_{r}^{stu}z̃_{s}z̃_{t}z̃_{u} + …,

where the linear coefficients form block matrices A = (a_{r}^{s}) and A^{−1} = (b_{r}^{s}), fixing the tangent space as described above.

The six real defining equations of M in ℂ^{5} can be expressed in the z_{s} coordinates as a graph, over a region in the tangent space:

M = {z⃗ : y_{2} = H_{2}, y_{3} = H_{3}, z_{4} = h_{4}, z_{5} = h_{5}}.

The functions H_{2}(z_{1}, z̄_{1}, x_{2}, x_{3}) and H_{3}(z_{1}, z̄_{1}, x_{2}, x_{3}) are real-valued, the functions h_{4}(z_{1}, z̄_{1}, x_{2}, x_{3}) and h_{5}(z_{1}, z̄_{1}, x_{2}, x_{3}) are complex-valued, and all are real analytic, vanishing to second order at the origin. For example, h_{5} begins with quadratic terms:

h_{5} = αz_{1}^{2} + βz̄_{1}^{2} + γz_{1}z̄_{1} + c^{ab}x_{a}x_{b} + d^{a}x_{a}z_{1} + e^{a}x_{a}z̄_{1} + O(3),

where the summations are over 2 ≤ a ≤ b ≤ 3 and O(3) denotes terms of degree three or higher. This expression can be simplified by a linear change of coordinates.

Lemma 2.1. If |β| ≠ |γ|/2, then there exist complex numbers a_{1}^{2} and a_{1}^{3}, such that in new variables z̃_{1} = z_{1} + a_{1}^{2}z_{2} + a_{1}^{3}z_{3}, z̃_{s} = z_{s} for s = 2, …, 5, the above defining function h_{5} is transformed to:

z̃_{5} = αz̃_{1}^{2} + β¯z̃_{1}^{2} + γz̃_{1} ¯z̃_{1} + c̃^{ab}x̃_{a}x̃_{b} + d̃^{a}x̃_{a}z̃_{1} + O(3).

This Lemma appears as part of the calculations of [Bishop], so a proof is not given here. Note that the coefficients c^{ab} and d^{a} may change, the e^{a}x_{a}z̄_{1} terms are eliminated, and the coefficients α, β, and γ are unchanged. The coordinate change may also change some coefficients in h_{5} of the higherorder terms, and of some of the terms appearing in the other three defining equations. Also note that no holomorphic coordinate change of the form z̃_{5} = z_{5} + p_{5}^{st}z_{s}z_{t} could eliminate the e^{a}x_{a}z̄_{1} terms; transforming the z_{1} variable is necessary, and under the hypothesis of the Lemma, the e^{2}, e^{3} coefficients can be transformed to any pair of complex numbers. The idea of the Lemma, using a holomorphic transformation of z_{1} to eliminate terms involving z̄_{1}, will be generalized to be the key idea for the solution of the normal form problem (Theorem 3.16).

2.2. The quadratic normal form. The general form of the real functions H_{2}, H_{3}, and complex functions h_{4}, h_{5}, which define M in ℂ^{5}, is as follows:

z_{2} = x_{2} + iH_{2} = x_{2} + i(Re(k_{2}^{a}x_{a}z_{1} + k_{2}z_{1}^{2}) + r_{2}^{ab}x_{a}x_{b} + r_{2}z_{1}z̄_{1}) + O(3)
z_{3} = x_{3} + iH_{3} = x_{3} + i(Re(k_{3}^{a}x_{a}z_{1} + k_{3}z_{1}^{2}) + r_{3}^{ab}x_{a}x_{b} + r_{3}z_{1}z̄_{1}) + O(3)
z_{4} = h_{4} = α_{4}z_{1}^{2} + β_{4}z̄_{1}^{2} + γ_{4}z_{1}z̄_{1} + c_{4}^{ab}x_{a}x_{b} + d_{4}^{a}x_{a}z_{1} + e_{4}^{a}x_{a}z̄_{1} + O(3)
z_{5} = h_{5} = α_{5}z_{1}^{2} + β_{5}z̄_{1}^{2} + γ_{5}z_{1}z̄_{1} + c_{5}^{ab}x_{a}x_{b} + d_{5}^{a}x_{a}z_{1} + e_{5}^{a}x_{a}z̄_{1} + O(3).

The coefficients are all complex except for the real coefficients r on the “conjugation-invariant” terms in H_{2} and H_{3}.

The plan is to find a coordinate change (1 − 2) to convert these quadratic quantities into a simpler “normal form.” The linear and secondorder coordinate changes that affect the quadratic terms are of the form z̃_{r} = a_{r}^{d}z_{d} + p_{r}^{st}z_{s}z_{t} for 1 ≤ d ≤ 5 and 1 ≤ s ≤ t ≤ 3. The linear coefficients a_{1}^{4}, a_{1}^{5} and the nonlinear coefficients p_{1}^{st} do not contribute quadratic terms.

The “first non-degeneracy condition” is that the matrix of coefficients

( \begin{matrix}β_{4} & γ_{4} \\ β_{5} & γ_{5} \end{matrix} ) is nonsingular. If the matrix has rank 1 or 0, this degeneracy

is preserved under any change of coordinates, and the degenerate normal forms will be listed in the next Subsection. The case of main interest is that the matrix has rank 2, so that the block ( \begin{matrix}a_{4}^{4} & a_{4}^{5} \\ a_{5}^{4} & a_{5}^{5} \end{matrix} ) can be used to

normalize these coefficients to ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ). Then Lemma 2.1 applies to h_{5} to

eliminate the e_{5}^{a}x_{a}z̄_{1} terms, using the linear coefficients a_{1}^{2}, a_{1}^{3}.

Lemma 2.1 cannot be used again, on the h_{4} terms, without possibly reintroducing x_{a}z̄_{1} terms into h_{5}. (This problem will occur later, in the elimination of higher-order terms from both h_{4} and h_{5}; the matter will be resolved by transforming the x_{a} variables to normalize h_{4}.) The current state of the quadratic terms is:

h_{4} = z̄_{1}^{2} + α_{4}z_{1}^{2} + c_{4}^{ab}x_{a}x_{b} + d_{4}^{a}x_{a}z_{1} + e_{4}^{a}x_{a}z̄_{1} + O(3)
h_{5} = z_{1}z̄_{1} + α_{5}z_{1}^{2} + c_{5}^{ab}x_{a}x_{b} + d_{5}^{a}x_{a}z_{1} + O(3).

The sum e_{4}^{a}x_{a}z̄_{1} can be rewritten as a matrix product:

( 1 i ) ( \begin{matrix}Re(e_{4}^{2}) & Re(e_{4}^{3}) \\ Im(e_{4}^{2}) & Im(e_{4}^{3}) \end{matrix} ) ( \begin{matrix}x_{2} \\ x_{3} \end{matrix} ) z̄_{1},

and the “second non-degeneracy condition” is that this real 2 × 2 coefficient matrix is non-singular. In this case, the real block R can transform the x_{2}, x_{3} coordinates so that these terms have the form x_{2}z̄_{1} + ix_{3}z̄_{1}. The remaining quadratic terms, for example, d_{4}^{2}x_{2}z_{1}, do not involve z̄_{1}, and their coefficients can be made to vanish, or to attain any complex value, by the non-linear transformations of the form z̃_{4} = z_{4} + p_{4}^{12}z_{1}z_{2}, etc. Quadratic transformations of z_{2} and z_{3} can eliminate most of the terms of H_{2} and H_{3}, except the z_{1}z̄_{1} terms. These terms, assuming the first non-degeneracy condition holds, can be eliminated by linear transformations of the form z̃_{2} = z_{2} + a_{2}^{5}z_{5}, z̃_{3} = z_{3} + a_{3}^{5}z_{5}.

With as many terms as possible eliminated, the non-degenerate quadratic normal form for M is:

{y_{2} = O(3), y_{3} = O(3), z_{4} = z̄_{1}^{2} + x_{2}z̄_{1} + ix_{3}z̄_{1} + O(3), z_{5} = |z_{1}|^{2} + O(3)}.

To make a normal form more convenient for later calculations, “complete the square” for h_{4}, and use the same linear factor L̄ = z̄_{1} + x_{2} + ix_{3} in h_{5}:

h_{4}(z_{1}, z̄_{1}, x_{2}, x_{3}) = (z̄_{1} + x_{2} + ix_{3})^{2} + O(3) = L̄^{2} + O(3)
h_{5}(z_{1}, z̄_{1}, x_{2}, x_{3}) = z_{1}(z̄_{1} + x_{2} + ix_{3}) + O(3) = z_{1}L̄ + O(3).

If the linear coordinate change A is required to fix an orientation of the real tangent plane, then the second non-degeneracy condition leads to two alternatives for the new coefficients on x_{a}z̄_{1} terms: either x_{2}z̄_{1} + ix_{3}z̄_{1} or x_{2}z̄_{1} − ix_{3}z̄_{1} in the first set of equations, or L̄ = z̄_{1} + x_{2} ± ix_{3} in the second set. The ± sign is not a biholomorphic invariant since it can change under the previously mentioned transformation z̃_{3} = −z_{3}. The two normal forms are also related by complex conjugation of all five coordinates.

2.3. Degenerate cases of the normal form. Complex tangents where the non-degeneracy conditions fail can still be put into a quadratic normal form, with the possibility of continuous invariants. The quadratic terms of the real-valued functions H_{2} and H_{3} can be made to vanish, or, if there is no z_{1}z̄_{1} term in h_{4} or h_{5}, then H_{2} = r_{2}z_{1}z̄_{1} +O(3) and H_{3} = r_{3}z_{1}z̄_{1} +O(3). The quadratic normal forms (in the unoriented case) for h_{4} and h_{5}, including the full rank case, are as follows:

h_{4} = (z̄_{1} + e_{4}^{2}x_{2} + e_{4}^{3}x_{3})^{2} + O(3),
h_{5} = z_{1}(z̄_{1} + e_{4}^{2}x_{2} + e_{4}^{3}x_{3}) + O(3), (e_{4}^{2}, e_{4}^{3}) ∈ {(1, i), (1, 0), (0, 0)};

h_{4} = β(z_{1}^{2} + z̄_{1}^{2}) + |z_{1}|^{2} + O(3), β ∈ [0, \frac{1}{2}) ∪ (\frac{1}{2}, ∞],
h_{5} = e_{5}^{2}x_{2}z̄_{1} + e_{5}^{3}x_{3}z̄_{1} + O(3), (e_{5}^{2}, e_{5}^{3}) ∈ {(1, i), (1, 0), (0, 0)};

h_{4} = e_{4}^{2}x_{2}z̄_{1} + \frac{1}{2}(z_{1} + z̄_{1})^{2} + O(3),
h_{5} = e_{5}^{2}x_{2}z̄_{1} + e_{5}^{3}x_{3}z̄_{1} + O(3),
(e_{4}^{2}, e_{5}^{2}, e_{5}^{3}) ∈ {(1, 0, 1), (1, 0, 0), (0, 1, i), (0, 1, 0), (0, 0, 0)};
h_{4} = e_{4}^{2}x_{2}z̄_{1} + O(3),
h_{5} = e_{5}^{3}x_{3}z̄_{1} + O(3), (e_{4}^{2}, e_{5}^{3}) ∈ {(1, 1), (1, 0), (0, 0)}.

For example, the defining functions z_{4} = L̄^{2}, z_{5} = LL̄ = (z_{1} + x_{2} − ix_{3})(z̄_{1} + x_{2} + ix_{3}) satisfy the first non-degeneracy condition, but not the second, and can be transformed into the (0, 0) case of the first group.

The h_{4} functions in the second and third groups resemble the “elliptic”, “hyperbolic”, and “parabolic” cases when m = n. If M^{2} = {z_{2} = h(z_{1}, z̄_{1})} is a surface in ℂ^{2} with a complex tangent, then M^{2} ×ℝ^{2} ×{0} = {y_{2} = y_{3} = z_{5} = 0, z_{4} = h(z_{1}, z̄_{1})}, contained inside ℂ^{4} ⊆ ℂ^{5} is not only degenerate (in the sense that the first non-degeneracy condition is not satisfied), but the locus of complex tangents is two-dimensional.

More generally, it can be shown that if a real manifold M has order of contact higher than 2 with any smooth complex hypersurface, then the first non-degeneracy condition fails.

However, in a neighborhood of a non-degenerate CR singularity, M is contained in a singular complex hypersurface in ℂ^{5}, which could be considered as a complexification of M. The normal form variety M̃^{4}, defined by y_{2} = y_{3} = 0, z_{4} = L̄^{2}, z_{5} = z_{1}L̄, is contained in the hypersurface H defined by z_{1}^{2}z_{4} − z_{5}^{2} = 0. Algebraically, this equation is obtained by eliminating the z̄_{1} variable. Geometrically, this hypersurface can be obtained by replacing the z̄ variables in the defining equations for M̃^{4} by new variables w (see Section 5, and [W_{84}]), to get a complex 4-submanifold of ℂ^{10}, and then projecting this smooth manifold into the original ℂ^{5} so that its image is the singular variety H. The containment of M̃^{4} inside H exhibits some interesting geometry (briefly considered in [C] and possibly also in some future paper), and resembles the parametrized cross-cap (u^{2}, v, uv) inside the Whitney umbrella variety xy^{2} − z^{2} = 0.

2.4. m-manifolds in ℂ^{n}, m < n. Neglecting the orientation issue, and using the L̄ = z̄_{1} + x_{2} + ix_{3} abbreviation, the generic complex tangent for m-manifolds M in ℂ^{n}, for \frac{2}{3}(n+1) ≤ m < n, has a normal form generalizing the m = 4, n = 5 case. The tangent space has exactly one complex direction, with coordinate z_{1}, and m − 2 real coordinates x_{2}, …, x_{m−1}. The 2n − mnormal space has exactly n − m + 1 complex directions, z_{m}, …, z_{n}, and m − 2 remaining real coordinates, y_{2}, …, y_{m−1}. The graphing functions (H, h) can be put into the form:

y_{a=2,...,m−1} = O(3)
z_{t=m,...,n−2} = (x_{4+2(t−m)} + ix_{5+2(t−m)})L̄ + O(3)
z_{n−1} = L̄^{2} + O(3)
z_{n} = z_{1}L̄ + O(3).

This normal form again assumes that two non-degeneracy conditions on the h functions are satisfied, the first being a full rank condition on the z̄_{1}^{2} and z_{1}z̄_{1} coefficients, and the second being a full rank condition on the real and imaginary parts of the x^{a}z̄_{1} coefficients, a = 2, …, m − 1, in the equations h_{t}, t = m, …, n − 2, and h_{n−1}. The indices on the z_{t} terms range from z_{m} = (x_{4} + ix_{5})L̄ + O(3) to z_{n−2} = (x_{2(n−m)} + ix_{2(n−m)+1})L̄ + O(3). This indexing scheme breaks down when m < \frac{2}{3}(n + 1), but this is outside the topological stability range under consideration. When m = n − 1, the functions z_{t} = h_{t} in this normal form do not appear. When m = \frac{2}{3}(n + 1), the complex tangents are isolated, but if m > \frac{2}{3}(n + 1), the 3m − 2n − 2 variables x_{2(n−m+1)}, …, x_{m−1} do not appear in the above quadratic terms, and in fact 3m−2n−2 is the real dimension of the locus N ⊆ M of complex tangents. When N has positive dimension, the quadratic normal form has the structure of a product of a totally real plane and a submanifold with an isolated complex tangent. This geometry is comparable to the normal forms of [Wh_{58}] and [H].

3. A linearized CR normal form problem

3.1. The nonlinear functional equation. Finding a formal or analytic transformation bringing a CR singularity of M in ℂ^{5} to normal form requires solving a system of functional equations. Starting with the non-degenerate normal form,

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

where H_{2}, H_{3}, e_{4}, e_{5} are vanishing to third order, the goal is to find a (formal) holomorphic coordinate transformation of the form (1), with identity linear part:

(3)  z̃_{s} = z_{s} + p_{s}(z_{1}, z_{2}, z_{3}, z_{4}, z_{5})

for s = 1, …, 5, so that the defining functions in the new coordinates are

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

Notation 3.1. Define E to be the real vector space of quadruples of formal power series e⃗ = (H_{2}, H_{3}, e_{4}, e_{5}), where H_{2}, H_{3} are formally real-valued, with terms of degree at least 2, and e_{4}, e_{5} are complex-valued, with terms of degree at least 3. Define P to be the complex vector space of quintuples p⃗ = (p_{1}, …, p_{5}) of complex-valued formal power series in z_{1}, …, z_{5} without constant terms.

A normalizing transformation exists, if given e⃗ = (H_{2}, H_{3}, e_{4}, e_{5}) ∈ E, there exists a solution p⃗ ∈ P of the following four equations:

0 = Im(z̃_{2}) = Im(z_{2} + p_{2}(z⃗))
(4)  = H_{2}(z_{1}, z̄_{1}, x_{2}, x_{3}) + Im(p_{2})
0 = Im(z̃_{3}) = Im(z_{3} + p_{3}(z⃗))
(5)  = H_{3}(z_{1}, z̄_{1}, x_{2}, x_{3}) + Im(p_{3})
0 = z̃_{4} − \overline{L̃}^{2}
= h_{4}(z_{1}, z̄_{1}, x_{2}, x_{3}) + p_{4}(z⃗)
−(z̄_{1} + x_{2} + ix_{3} + \overline{p_{1}(z⃗)} + Re(p_{2}(z⃗)) + iRe(p_{3}(z⃗)))^{2}
(6)  = e_{4}(z_{1}, z̄_{1}, x_{2}, x_{3}) + p_{4} − 2L̄(\overline{p_{1}} + Re(p_{2}) +iRe(p_{3}))
−(\overline{p_{1}} + Re(p_{2}) + iRe(p_{3}))^{2}
0 = z̃_{5} − z̃_{1}\overline{L̃}
= h_{5}(z_{1}, z̄_{1}, x_{2}, x_{3}) + p_{5}(z⃗)
−(z_{1} + p_{1}(z⃗))(L̄ + \overline{p_{1}(z⃗)} + Re(p_{2}(z⃗)) +iRe(p_{3}(z⃗)))
(7)  = e_{5}(z_{1}, z̄_{1}, x_{2}, x_{3}) + p_{5} − L̄p_{1} − z_{1}(\overline{p_{1}} + Re(p_{2}) + iRe(p_{3}))
−p_{1}(\overline{p_{1}} + Re(p_{2}) + iRe(p_{3})),

where the functions p_{1}, … p_{5} appearing in Equations (4 − 7) are restricted to the points z⃗ = (z_{1}, x_{2} + iH_{2}, x_{3} + iH_{3}, h_{4}, h_{5}) on M. The action of the coordinate change (3) on the series e⃗ can be described as a non-linear map F : E × P → E, taking the pair (e⃗, p⃗) to the quantities (4 − 7).

The analytic normal form problem is to start with e⃗ convergent on some polydisc {|z_{1}| < r_{1}, |x_{2}| < r_{2}, |x_{3}| < r_{3}}, and to find p⃗ convergent on some polydisc {|z_{1}| < ρ_{1}, |z_{2}| < ρ_{2}, …, |z_{5}| < ρ_{5}} so that F (e⃗, p⃗) = 0_{E}. If p⃗ is not an exact solution, F (e⃗, p⃗) will have to be converted into the z̃ variables, using the inverse transformation (2), z = A^{−1}z̃+q(z̃), to get the higher-order terms in the defining functions in the z̃ coordinate system. 3.2. The stabilizer of the polynomial normal form. Recall M̃^{4} denotes the 4-manifold already exactly defined by quadratic and linear polynomials, with H_{2}, H_{3}, e_{4}, e_{5} all identically zero. There are some coordinate transformations (3) which preserve the form of the defining equations, for example, if the five functions p_{s} all vanish on M̃^{4}. However, these are not the only such transformations in the “stabilizer” of the variety, defined to be the set of formal transformations (3) with p⃗ ∈ P such that F (0_{E}, p⃗) = 0_{E}.

Substituting e⃗ = 0_{E} into Equations (4 − 7) gives

0 = Im(p_{2}) = Im(p_{3})
0 = p_{4} − 2L̄(\overline{p_{1}} + Re(p_{2}) + iRe(p_{3})) − (\overline{p_{1}} + Re(p_{2}) + iRe(p_{3}))^{2}
0 = p_{5} − L̄p_{1} − z_{1}(\overline{p_{1}} + Re(p_{2}) + iRe(p_{3})) − p_{1}(\overline{p_{1}}+Re(p_{2}) + iRe(p_{3})),

where all the functions p_{s} are restricted to M̃^{4}. Since z_{4} = L̄^{2} and z_{5} = z_{1}L̄ are related by the expression z_{1}^{2}z_{4} = z_{5}^{2}, any formal power series p_{s}(z⃗) in the ideal J ⊆ ℂ[[z_{1}, …, z_{5}]] generated by z_{1}^{2}z_{4} − z_{5}^{2} is zero when restricted to z_{4} = L̄^{2} and z_{5} = z_{1}L̄. Recall J defines a singular complex hypersurface H in ℂ^{5} containing M̃^{4}.

The first two equations imply p_{2} and p_{3} must be real-valued on M̃^{4}, and the second two,

(8) p_{4}(z_{1}, x_{2}, x_{3}, L̄^{2}, z_{1}L̄) = (2L̄ + \overline{p_{1}} + p_{2} + ip_{3})(\overline{p_{1}} + p_{2} + ip_{3})
(9) p_{5}(z_{1}, x_{2}, x_{3}, L̄^{2}, z_{1}L̄) = L̄p_{1} + (z_{1} + p_{1})(\overline{p_{1}} + p_{2} + ip_{3}),

show that p_{4} and p_{5} are determined by p_{1}, p_{2}, p_{3} in ℂ[[z_{1}, …, z_{5}]]/J, but p_{1}, p_{2}, p_{3} cannot be arbitrary. For example, if p_{2} = rx_{2}^{a}x_{3}^{b}, with r real, then 2rL̄x_{2}^{a}x_{3}^{b}+r^{2}x_{2}^{2a}x_{3}^{2b} is not a function of the arguments z_{1}, x_{2}, x_{3}, L̄^{2}, z_{1}L̄, and Equation (8) cannot hold for any p_{4} unless r = 0.

Notation 3.2. Let Q ⊆ ℂ[[z_{1}, L̄, x_{2}, x_{3}]] denote the subalgebra of formal power series which can be rewritten as series in z_{1}, x_{2}, x_{3}, L̄^{2}, z_{1}L̄.

The only monomials not in Q are those of the form x_{2}^{a}x_{3}^{b}L̄^{odd}. In particular, any expression f in z_{1}, z̄_{1}, x_{2}, x_{3}, satisfies z_{1} · f ∈ Q. This simplifies the conditions (8 − 9), so that p_{1}, p_{2}, p_{3} ∈ ℂ[[z_{1}, z_{2}, z_{3}, z_{4}, z_{5}]] are components of a stabilizing transformation if their restrictions to y_{2} = y_{3} = 0, z_{4} = L̄^{2}, z_{5} = z_{1}L̄ satisfy:

\{2L̄(\overline{p_{1}} + p_{2} + ip_{3}) + \overline{p_{1}}(\overline{p_{1}} + 2(p_{2} + ip_{3})), (L̄ + \overline{p_{1}})p_{1}\} ⊆ Q.

It should be remarked that there are some linear transformations of ℂ^{5} which preserve the normal form variety, for example, scaling z_{1}, z_{2}, z_{3} by σ > 0 and z_{4} and z_{5} by σ^{2}, but these will not be considered here.

3.3. The linear problem. A formal solution of the non-linear problem F (e⃗, p⃗) = 0_{E} can be approximated by finding a solution p⃗ of a related linear equation. Such a p⃗ is an approximation in the sense that if the higher-order terms of the defining equations, e⃗, have lowest degree d ≥ 3, then the coordinate change (3) transforms the equations so that the lowest degree is approximately doubled. Iterating this procedure, and composing the coordinate transformations, determines as many terms in an exact formal solution as desired.

Equations (10 − 13) define a real linear map dF : P → E, formally a derivative of F (restricted to {0_{E}} × P) at the point (0_{E}, 0_{P}), acting on p⃗ ∈ P to give a quadruple (H_{2}, H_{3}, e_{4}, e_{5}) of power series in z_{1}, z̄_{1}, x_{2}, and x_{3}:

(10) H_{2} = Im(p_{2}(z_{1}, x_{2}, x_{3}, L̄^{2}, z_{1}L̄)),
(11) H_{3} = Im(p_{3}(z_{1}, x_{2}, x_{3}, L̄^{2}, z_{1}L̄)),
e_{4} = p_{4}(z_{1}, x_{2}, x_{3}, L̄^{2}, z_{1}L̄) − 2L̄\overline{p_{1}(z_{1}, x_{2}, x_{3}, L̄^{2}, z_{1}L̄)}

−2L̄(Re(p_{2}(z_{1}, x_{2}, x_{3}, L̄^{2}, z_{1}L̄)) + iRe(p_{3}(z_{1}, x_{2}, x_{3}, L̄^{2}, z_{1}L̄))), (12)

e_{5} = p_{5}(z_{1}, x_{2}, x_{3}, L̄^{2}, z_{1}L̄) − L̄p_{1}(z_{1}, x_{2}, x_{3}, L̄^{2}, z_{1}L̄)

−z_{1}\overline{p_{1}(z_{1}, x_{2}, x_{3}, L̄^{2}, z_{1}L̄)} (13)

−z_{1}(Re(p_{2}(z_{1}, x_{2}, x_{3}, L̄^{2}, z_{1}L̄)) + iRe(p_{3}(z_{1}, x_{2}, x_{3}, L̄^{2}, z_{1}L̄))).

This linearization idea was used by [M] in another CR normal form problem, but with one equation instead of a system of equations. Note that the output involves conjugation of functions of z_{1} and L̄, so some rearrangement may be needed to express the output as power series in E (variables z_{1}, z̄_{1}, x_{2}, x_{3}) or as power series in z_{1}, L̄, x_{2}, x_{3}. Also, the restriction of all the functions p_{s} to the values z_{2} = x_{2}, z_{3} = x_{3}, z_{4} = L̄^{2}, z_{5} = z_{1}L̄ motivates a definition of the weight of their terms:

Definition 3.3. The degree of a monomial z_{1}^{a_{1}}z̄_{1}^{a_{2}}x_{2}^{a_{3}}x_{3}^{a_{4}} or z_{1}^{a_{1}} L̄^{a_{2}}x_{2}^{a_{3}}x_{3}^{a_{4}} is the integer a_{1} + a_{2} + a_{3} + a_{4}, and the order of a power series refers to the smallest among the degrees of its monomials (with non-zero coefficients). The weight of a monomial z_{1}^{a_{1}}z_{2}^{a_{2}}z_{3}^{a_{3}}z_{4}^{a_{4}}z_{5}^{a_{5}} is the integer a_{1} + a_{2} + a_{3} + 2a_{4} + 2a_{5}, and the weight of a power series is the smallest of its monomials’ weights.

If (H_{2}, H_{3}, e_{4}, e_{5}) are power series of order d − 1, d − 1, d, d, with d ≥ 3, then the power series (p_{1}, p_{2}, p_{3}, p_{4}, p_{5}) in a solution of (10−13) are of weight d − 1, d − 1, d − 1, d, and d. If p⃗ is a solution of the linear problem, then comparing (10 − 13) with (4 − 7) shows that the output F (e⃗, p⃗) of the nonlinear function is four power series with order 2d − 3 for the new H_{2} and H_{3}, and with order 2d − 2 for the new e_{4} and e_{5}.

The existence of stabilizing transformations corresponds to a non-trivial kernel S ⊆ P, defined by equations similar to those of the stabilizer: p_{2} and p_{3} must be real-valued on M̃^{4}, and

p_{4}(z_{1}, x_{2}, x_{3}, L̄^{2}, z_{1}L̄) = 2L̄(\overline{p_{1}} + p_{2} + ip_{3})
p_{5}(z_{1}, x_{2}, x_{3}, L̄^{2}, z_{1}L̄) = L̄p_{1} + z_{1}(\overline{p_{1}} + p_{2} + ip_{3}),

which determines p_{4} and p_{5} mod J, and imposes on p_{1}, p_{2}, p_{3} the condition that their restrictions to M̃^{4} satisfy \{L̄(\overline{p_{1}} + p_{2} + ip_{3}), L̄p_{1}\} ⊆ Q.

3.4. Normalized transformations. Choosing a “normalization” for the solution of the problem dF (p⃗) = e⃗ corresponds to choosing a subspace C of P complementary to the kernel S. To prove the claim that dF is surjective, the approach will be to start with e⃗ and construct an element p⃗ ∈ C.

Motivated by the defining conditions for the kernel, the following normalization is considered for the transformation functions p_{s}. The idea is to decompose each p_{s} into a sum of functions of fewer than five variables:

(14)  p_{1} = z_{1}^{2}p_{1}^{H}((z_{1} + z_{2} − iz_{3})^{2}, z_{2}, z_{3}) + p_{1}^{A}(z_{2}, z_{3}, z_{4})
(15)  p_{a} = p_{a}^{R}(z_{2}, z_{3}) + p_{a}^{I}(z_{1}, z_{2}, z_{3}, z_{4}, z_{5})
(16)  p_{4} = p_{4}^{E}(z_{1}, z_{2}, z_{3}, z_{4}) + z_{5}p_{4}^{O}(z_{1}, z_{2}, z_{3}, z_{4})
(17)  p_{5} = p_{5}^{E}(z_{1}, z_{2}, z_{3}, z_{4}) + z_{5}p_{5}^{O}(z_{1}, z_{2}, z_{3}, z_{4}),

where for a = 2, 3, p_{a}^{R}(z_{2}, z_{3}) has real coefficients. The terms p_{a}^{I} are of the following form:

(18) p_{a}^{I} = p_{a}^{E}(z_{1}, z_{2}, z_{3}, z_{4}) + z_{5}p_{a}^{O}(z_{1}, z_{2}, z_{3}, z_{4}) + p′_{a}(z_{1}, z_{2}, z_{3})

with p_{a}^{E} + z_{5}p_{a}^{O} being (formally) purely imaginary when restricted to M̃^{4}, and p′_{a}(z_{1}, x_{2}, x_{3}) equal to a sum of monomials L^{odd}x_{2}^{a}x_{3}^{b} with complex coefficients. In some sense, the conditions on the functions p_{1}, p_{2}, p_{3} are complementary to the conditions defining the kernel. The p^{E}(z_{1}, z_{2}, z_{3}, z_{4}) + z_{5}p^{O}(z_{1}, z_{2}, z_{3}, z_{4}) decomposition in (16 − 18) defines a vector space complement of J in ℂ[[z_{1}, …, z_{5}]], formed by remainders r in applying the Division Theorem to p = (z_{5}^{2} + z_{1}^{2}z_{4})q + r.

The capital superscripts are not multiindices but rather denote the roles of each term in the solution. The p_{a}^{R} is a function of “real” variables, and the “imaginary” part of p_{a}^{I} will eliminate the H_{a} functions. The p_{4} and p_{5} functions, when decomposed into parts p^{E} and p^{O}, have “even” and “odd” powers of L̄, and can eliminate all the Q monomials in the h_{4} and h_{5} functions. The remaining monomials are of the form x_{2}^{a}x_{3}^{b}L̄^{odd} in h_{4} and h_{5}, and can be handled in the same way as in Lemma 2.1. p_{1}^{H}, a “holomorphic” function in z_{1}, eliminates non-Q monomials when conjugated in (12), and p_{1}^{A} becomes an “antiholomorphic” expression when z_{4} is equal to L̄^{2}, eliminating non-Q monomials in (13). The z_{1}^{2} factor in z_{1}^{2}p_{1}^{H} means it has no monomials in common with p_{1}^{A}, and it is in Q when restricted to M̃^{4}. Without this z_{1}^{2}, p_{1}^{H} and p_{1}^{A} would both introduce x_{2}^{a}x_{3}^{b}L̄^{1} terms in h_{4} and h_{5}; this is why Lemma 2.1 could not be used twice on the quadratic terms. Although p_{1}, as normalized, can eliminate x_{2}^{a}x_{3}^{b}L̄^{1} terms only from h_{5}, these terms can be eliminated from h_{4} by the L̄(p_{2}^{R} + ip_{3}^{R}) appearing in (12), without re-introducing any in (13).

Supposing that e⃗ is of the form where H_{2} and H_{3} are homogeneous of degree d − 1, and e_{4} and e_{5} are homogeneous of degree d, the normalized solution of dF (p⃗) = e⃗ should have p_{1}^{H} with degree d − 3, p_{1}^{A}, p_{2}^{R}, p_{3}^{R}, p_{2}^{I}, p_{3}^{I} with weight d − 1, p_{4}^{E}, and p_{5}^{E} with weight d, and p_{4}^{O} and p_{5}^{O} with weight d − 2. It can be checked that such polynomials form a real vector subspace of P exactly equal in (real) dimension to the subspace of E defined by homogeneous polynomials with the stated degrees.

It is convenient to think of the transformation (3) having “identity linear part,” but if the function h_{4} contains a cubic term z̄_{1}^{3}, it can only be eliminated by a linear transformation of the form z̃_{1} = z_{1} + a_{1}^{4}z_{4}, which could be considered as a term in p_{1}^{A} of weight 2. This is the only exception that needs to be made when thinking of the functions p_{s} as order ≥ 2.

Inspecting the form of Equations (10−13), each p_{s} function appears with terms involving L̄, but z̄_{1} also appears by itself (not collected into L̄ factors), when the p_{1}, p_{2}, and p_{3} functions undergo complex conjugation. The normalization for p_{1}^{H}, as a power series in z_{2}, z_{3}, and L^{2}, is chosen so that L̄^{2} appears in \overline{p_{1}^{H}}. At this point, it seems natural to rearrange the e⃗ components as power series in z_{1}, L̄, x_{2}, x_{3}, so that like terms can be compared on both sides of Equations (10 − 13). However, this rearrangement has serious consequences; if the rearranged series is decomposed into a sum, and each subseries expanded back into the original variables, the domain can shrink significantly. Nonetheless, the strategy for constructing an inverse of dF : C → E will be to rearrange e⃗, and define the p_{s} series using the coefficients of the rearranged e⃗. The p_{s} series will be shown to converge on a small polydisc, using a Banach space norm suited to this type of construction.

3.5. Preliminary lemmas on power series and change of variables. Some standard notions ([GF], [Walter]) about power series in real and complex variables will be used in analyzing the functional equation.

Notation 3.4. For r = (r_{1}, …, r_{n}) ∈ ℝ^{n}, with all r_{k} > 0, define a polydisc in ℂ^{n} by 𝔻_{r} = {(z_{1}, …, z_{n}) : |z_{k}| < r_{k}}.

Notation 3.5. For multi-indexed formal power series centered at 0 ∈ ℂ^{n}, c(z_{1}, …, z_{n}) = Σc^{α}z^{α} = Σc^{α_{1}...α_{n}}z_{1}^{α_{1}} … z_{n}^{α_{n}}, c^{α} ∈ ℂ, define

|c|_{r} = Σ|c^{α}|r^{α} = Σ|c^{α_{1}...α_{n}}|r_{1}^{α_{1}} … r_{n}^{α_{n}}.

Lemma 3.6. If |c|_{r} < ∞ then Σc^{α}z^{α} converges absolutely for all z ∈ 𝔻_{r}, and c(z_{1}, …, z_{n}) defines a bounded, holomorphic function on 𝔻_{r}. The set of formal power series c such that |c|_{r} < ∞ is a complex (and real) Banach space.

Lemma 3.7. If |c|_{r} < ∞, then any power series d = Σd^{α}z^{α} with |d^{α}| ≤ |c^{α}| for each α satisfies |d|_{r} ≤ |c|_{r}.

For example, this Lemma includes subseries d of c, meaning d^{α} = c^{α} for some indices and d^{α} = 0 for the rest.

Lemma 3.8. |z_{1} · c|_{r} = r_{1}|c|_{r}, and in general |b · c|_{r} ≤ |b|_{r} · |c|_{r}.

Notation 3.9. In ℂ × ℝ^{2}, define the polydisc D_{R} = {(z_{1}, x_{2}, x_{3}) : |z_{1}| < R, |x_{2}| < R, |x_{3}| < R}, and let A be the space of formal power series in z_{1}, z̄_{1}, x_{2}, x_{3} with complex coefficients. Define A_{R} to be the subspace of power series c such that |c|_{R} := Σ|c^{abcd}|R^{a+b+c+d} < ∞.

Using the coefficients of c = Σc^{abcd}z_{1}^{a}z̄_{1}^{b}x_{2}^{c}x_{3}^{d} ∈ A_{R} as coefficients in a power series c(z_{1}, w, z_{2}, z_{3}) defines a complex analytic function on 𝔻_{(R,R,R,R)} in ℂ^{4}, by Lemma 3.6. The restriction of this function to the real 4-plane {w = z̄_{1}, z_{2} = z̄_{2}, z_{3} = z̄_{3}} is a function whose power series c(z_{1}, z̄_{1}, x_{2}, x_{3}) is convergent on D_{R}. In particular, (A_{R}, | |_{R}) is a Banach algebra.

Lemma 3.10. If c = Σc^{abcd}z_{1}^{a}z̄_{1}^{b}x_{2}^{c}x_{3}^{d} ∈ A_{R}, then c(z_{1}, z̄_{1}, x_{2}, x_{3}) can be rearranged into a power series k(z_{1}, L̄, x_{2}, x_{3}) = Σk^{abcd}z_{1}^{a}(z̄_{1}+x_{2}+ix_{3})^{b}x_{2}^{c}x_{3}^{d}, where the coefficients k^{abcd} satisfy

(19)  Σ|k^{abcd}|3^{b}(R/9)^{a+b+c+d} ≤ |c(z_{1}, z̄_{1}, x_{2}, x_{3})|_{R},

and the sum Σk^{abcd}z_{1}^{a}L̄^{b}x_{2}^{c}x_{3}^{d} converges on the polydisc D_{R/9}.

Proof. The transformation from w to W = w+z_{2}+iz_{3} “shears” the polydisc 𝔻_{(R,R,R,R)} in ℂ^{4}. The norm |Σk^{abcd}z_{1}^{a}W^{b}x_{2}^{c}x_{3}^{d}|_{(R/9,R/3,R/9,R/9)} can be estimated in terms of the norm |c(z_{1}, z̄_{1}, x_{2}, x_{3})|_{R} by comparing the coefficients k^{abcd} to the coefficients c^{abcd}. Some trinomial coefficients appear, having the following properties ([Aigner]) for whole numbers b, j, k, l:

( \begin{matrix}b \\ jkl \end{matrix} ) = \{ \begin{matrix}\frac{b!}{j!k!l!} & if b = j + k + l \\ 0 & otherwise \end{matrix} \}; ∑_{j,k,l} ( \begin{matrix}b \\ jkl \end{matrix} ) = 3^{b}.

The degree n terms of the power series c can be expanded:

∑_{a+b+c+d=n} c^{abcd}z_{1}^{a}z̄_{1}^{b}x_{2}^{c}x_{3}^{d} = ∑_{a+b+c+d=n} c^{abcd}z_{1}^{a}(L̄ − x_{2} − ix_{3})^{b}x_{2}^{c}x_{3}^{d}

= ∑_{a+b+c+d=n} c^{abcd}z_{1}^{a}(∑_{j,k,l}(−1)^{j+k}i^{k}( \begin{matrix}b \\ jkl \end{matrix} )L̄^{l}x_{2}^{c+j}x_{3}^{d+k})

=  ∑_{a′+b′+c′+d′=n} k^{a′b′c′d′}z_{1}^{a′} L̄^{b′}x_{2}^{c′}x_{3}^{d′},

where the coefficients k are formed by collecting like terms:

k^{a′b′c′d′} = ∑_{\begin{matrix}b′ ≤ b \\ c ≤ c′ \\ d ≤ d′ \end{matrix}} c^{a′bcd}(−1)^{b−b′}i^{d′−d} ( \begin{matrix}b \\ c′ − c d′ − d b′ \end{matrix} ) .

By the triangle inequality, and then an interchange of summation, the degree n terms of the LHS of Inequality (19) sum to

∑_{a′+b′+c′+d′=n} |k^{a′b′c′d′}|\frac{3^{b′}R^{n}}{9^{n}} ≤ ∑_{a′+b′+c′+d′=n}( ∑_{\begin{matrix}b′ ≤ b \\ c ≤ c′ \\ d ≤ d′ \end{matrix}} |c^{a′bcd}|( \begin{matrix}b \\ c′ − c d′ − d b′ \end{matrix} ))\frac{3^{b′}R^{n}}{9^{n}}

≤ ∑_{a+b+c+d=n} |c^{abcd}|( ∑_{\begin{matrix}b′ ≤ b \\ c ≤ c′ \\ d ≤ d′ \end{matrix}} ( \begin{matrix}b \\ c′ − c d′ − d b′ \end{matrix} ) 3^{b′})\frac{R^{n}}{9^{n}}

≤ ∑_{a+b+c+d=n} |c^{abcd}|( ∑_{b′+c′+d′=n−a}( \begin{matrix}b \\ c′ − c d′ − d b′ \end{matrix} ))\frac{3^{b}R^{n}}{9^{n}}

≤ ∑_{a+b+c+d=n}|c^{abcd}|3^{b} · 3^{b} \frac{R^{n}}{9^{n}} ≤ ∑_{a+b+c+d=n} |c^{abcd}|R^{n},

and the absolutely convergent sum of these collections over all n is independent of the order of terms and gives the inequality. This argument actually shows that Σk^{abcd}z_{1}^{a}W^{b}z_{2}^{c}z_{3}^{d} has finite (R, R/3, R/9, R/9) norm, but the intersection of the polydisc 𝔻_{(R,R/3,R/9,R/9)} and the real 4-plane {W = z̄_{1} + x_{2} + ix_{3}, z_{2} = x_{2}, z_{3} = x_{3}} contains D_{R/9}, so the (R/9, R/3, R/9, R/9) norm is considered instead.

Notation 3.11. For formal power series in the variables z_{1}, L̄, x_{2}, x_{3}, define | |′_{r} by

|Σk^{abcd}z_{1}^{a}L̄^{b}x_{2}^{c}x_{3}^{d}|′_{r} = Σ|k^{abcd}|3^{b}r^{a+b+c+d}.

This is the r = (r, 3r, r, r) norm on the complex polydisc 𝔻_{r}.

If |k|′_{r} < ∞, then k(z_{1}, L̄, x_{2}, x_{3}) converges for (z_{1}, x_{2}, x_{3}) ∈ D_{r}, because |L̄| ≤ |z̄_{1}| + |x_{2}| + |x_{3}| < 3r. Lemma 3.10 can be restated for the rearrangement k(z_{1}, L̄, x_{2}, x_{3}) of c(z_{1}, z̄_{1}, x_{2}, x_{3}) in terms of this norm: |k|′_{R/9} ≤ |c|_{R}. The polyradius (R/9, R/3, R/9, R/9) chosen in Lemma 3.10 is convenient for calculations, but also is small enough so that using the Weierstrass Division Theorem as a step in the main Theorem will not further shrink the domain.

Note that the “prime” notation will always signify the presence of the “rearranged” variable L̄. A disadvantage of the | |′_{r} norm, and with the z_{1}, L̄, x_{2}, x_{3} coordinates in general, is the bad behavior with respect to complex conjugation. The conjugate of a power series k(z_{1}, L̄, x_{2}, x_{3}) in general requires a rearrangement to be expressed as another power series in the same variables. However, if a power series is independent of the holomorphic variable z_{1}, then its conjugate does not involve z̄_{1}. The following Lemma shows that the rearrangement of the conjugate does not increase the primed norm.

Lemma 3.12. If the power series k(z_{1}, L̄, x_{2}, x_{3}) = ∑ k^{0abc}L̄^{a}x_{2}^{b}x_{3}^{c} has finite | |′_{r} norm, and ∑ q^{a′0b′c′}z_{1}^{a′}x_{2}^{b′}x_{3}^{c′} is obtained by expanding L = z_{1} + x_{2} − ix_{3} in the conjugate expression k̄ = ∑ \overline{k^{0abc}}L^{a}x_{2}^{b}x_{3}^{c}, then |q|′_{r} ≤ |k|′_{r}.

Proof.

q^{a′0b′c′} = ∑_{\begin{matrix}a′ ≤ a \\ b ≤ b′ \\ c ≤ c′ \end{matrix}} \overline{k^{0abc}}(−i)^{c′−c} ( \begin{matrix}a \\ b′ − b c′ − c a′ \end{matrix} )

∑_{a′+b′+c′=n} |q^{a′0b′c′}|r^{n} ≤ ∑_{a+b+c=n} |k^{0abc}|( ∑_{\begin{matrix}a′ ≤ a \\ b ≤ b′ \\ c ≤ c′ \end{matrix}} ( \begin{matrix}a \\ b′ − b c′ − c a′ \end{matrix} ))r^{n}

≤ ∑_{a+b+c=n} |k^{0abc}|3^{a}r^{n}.

Notation 3.13. In analogy with Notation 3.1, let E′ denote the space of formally real-valued and complex-valued power series (H_{2}, H_{3}, e_{4}, e_{5}) in the variables z_{1}, L̄, x_{2}, x_{3}. E and E′ are isomorphic (as real vector spaces) by the rearrangement. Define E_{r} (and E′_{r}) to be the real vector subspace of E (E′) where all four components have order at least 3, and finite | |_{r} (| |′_{r}) norms, defining real-valued functions H_{2} and H_{3} and complex-valued functions e_{4} and e_{5} on D_{r}. The maximum norms are denoted

|e⃗|_{r} = max{|H_{2}|_{r}, |H_{3}|_{r}, |e_{4}|_{r}, |e_{5}|_{r}}; |e⃗|′_{r} = max{|H_{2}|′_{r}, |H_{3}|′_{r}, |e_{4}|′_{r}, |e_{5}|′_{r}},

so that (E_{r}, | |_{r}) and (E′_{r}, | |′_{r}) are Banach spaces.

By Lemma 3.10, rearrangement is a continuous, one-to-one map E_{R} → E′_{R/9}.

Notation 3.14. Recall P is the complex vector space of quintuples p⃗ of formal power series (p_{1}, p_{2}, p_{3}, p_{4}, p_{5}) in the variables z_{1}, …, z_{5}. Let Δ_{r} denote the polydisc 𝔻_{(r,r,r,9r^{2},3r^{2})} ⊆ ℂ^{5}. The r = (r, r, r, 9r^{2}, 3r^{2}) norm for power series p_{s}(z⃗) and the maximum norm for quintuples p⃗ = (p_{1}, …, p_{5}) are denoted

||p_{s}||_{r} = |Σp_{s}^{abcde}z_{1}^{a}z_{2}^{b}z_{3}^{c}z_{4}^{d}z_{5}^{e}|_{(r,r,r,9r^{2},3r^{2})} = Σ|p_{s}^{abcde}|9^{d}3^{e}r^{a+b+c+2d+2e},
||p⃗||_{r} = max_{s=1,...,5}{||p_{s}||_{r}}.

This superscript notation for the coefficients of p_{s} is different from that used in Section 2.

This polyradius is chosen so that a power series p_{s}, when restricted to M̃^{4}, defines a function of z_{1}, L̄, x_{2}, x_{3} with controlled norm:

|p_{s}(z_{1}, x_{2}, x_{3}, L̄^{2}, z_{1}L̄)|′_{r} = ||p_{s}(z_{1}, z_{2}, z_{3}, z_{4}, z_{5})||_{r}.

Notation 3.15. Define P_{r} as the subspace of P composed of quintuples p⃗ = (p_{1}, p_{2}, p_{3}, p_{4}, p_{5}) so that each p_{s} has weight at least 2, and the norm ||p⃗||_{r} is finite, so that (P_{r}, || ||_{r}) is a Banach space, and p⃗ ∈ P_{r} defines a complex analytic map Δ_{r} → ℂ^{5}.

3.6. A convergent solution, after the rearrangement. Recall the submanifold M was defined as the graph of real analytic functions H and h in the variables z_{1}, z̄_{1}, x_{2}, x_{3}, over a region in ℂ × ℝ^{2}, which can be assumed to contain a polydisc D_{R}. The following Theorem constructs the functions p⃗ approximating a solution of the normal form problem, and by estimating the norm of p⃗ in terms of the norm of e⃗, shows that p⃗ is convergent. However, p⃗ is defined in terms of the rearranged functions e(z_{1}, L̄, x_{2}, x_{3}), so its polyradius of convergence in ℂ^{5} is small compared to R.

Theorem 3.16. If e⃗ ∈ E_{R}, then there is a solution p⃗ of the equation dF (p⃗) = e⃗ such that p⃗ ∈ P_{R/9}. This p⃗ is the unique solution satisfying the normalization conditions.

Proof. Let r = R/9. By Lemma 3.10, the components of e⃗ can be rearranged to power series e⃗ = (H_{2}, H_{3}, e_{4}, e_{5}) ∈ E′_{r}, vanishing to third order in the rearranged variables, and converging on D_{r} = D_{R/9} ⊆ ℂ × ℝ^{2}. The idea is to find p⃗ = (p_{1}, p_{2}, p_{3}, p_{4}, p_{5}) by constructing the components p_{1}^{H}, …, in terms of e⃗. At one point, the Weierstrass Division Theorem is required, with estimates on quotients and remainders ([GF] §III.2.). The point of calculating these estimates is primarily to show that there is no further shrinking after the initial rearrangement.

The real-valued function H_{2}(z_{1}, L̄, x_{2}, x_{3}) can be decomposed into its Q and non-Q monomials:

H_{2} = ∑_{b even} H_{2}^{abcd}z_{1}^{a}L̄^{b}x_{2}^{c}x_{3}^{d} + ∑_{\begin{matrix}a > 0 \\ b odd \end{matrix}} H_{2}^{abcd}z_{1}^{a}L̄^{b}x_{2}^{c}x_{3}^{d} + ∑_{b odd} H_{2}^{0bcd}L̄^{b}x_{2}^{c}x_{3}^{d}.

Adding and subtracting the conjugate of the non-Q part gives two realvalued sums:

H_{2} = ∑_{b even} H_{2}^{abcd}z_{1}^{a}L̄^{b}x_{2}^{c}x_{3}^{d} + ∑_{\begin{matrix}a > 0 \\ b odd \end{matrix}} H_{2}^{abcd}z_{1}^{a}L̄^{b}x_{2}^{c}x_{3}^{d} − ∑_{b odd} \overline{H_{2}^{0bcd}}L^{b}x_{2}^{c}x_{3}^{d}

+ ∑_{b odd} \overline{H_{2}^{0bcd}}L^{b}x_{2}^{c}x_{3}^{d} + ∑_{b odd} H_{2}^{0bcd}L̄^{b}x_{2}^{c}x_{3}^{d}.

By Lemma 3.7, | ∑_{b odd} H_{2}^{0bcd}L̄^{b}x_{2}^{c}x_{3}^{d}|′_{r} ≤ |H_{2}|′_{r} ≤ |e⃗|′_{r}, and by Lemma 3.12,

if ∑_{b odd} \overline{H_{2}^{0bcd}}L^{b}x_{2}^{c}x_{3}^{d} is formally expanded as q_{2} = ∑ q_{2}^{bcd}z_{1}^{b}x_{2}^{c}x_{3}^{d}, then

|q_{2}(z_{1}, x_{2}, x_{3})|′_{r} ≤ | ∑_{b odd} H_{2}^{0bcd}L̄^{b}x_{2}^{c}x_{3}^{d}|′_{r}.

Define p_{2}^{I} = p′_{2} + p′′_{2} on Δ_{r}, by

p′_{2}(z_{1}, z_{2}, z_{3}) = 2i ∑ q_{2}^{bcd}z_{1}^{b}z_{2}^{c}z_{3}^{d},

p′′_{2} = i( ∑_{b even}H_{2}^{abcd}z_{1}^{a}z_{4}^{b/2}z_{2}^{c}z_{3}^{d}+z_{5} ∑_{\begin{matrix}a > 0 \\ b odd \end{matrix}}H_{2}^{abcd}z_{1}^{a−1}z_{4}^{(b−1)/2}z_{2}^{c}z_{3}^{d}−∑ q_{2}^{bcd}z_{1}^{b}z_{2}^{c}z_{3}^{d}).

Then, by Lemma 3.7 applied to the Q part of H_{2}, ||p′′_{2}||_{r} ≤ |H_{2}|′_{r} + |q_{2}|′_{r}. By construction, H_{2} = Im(p_{2}^{I}(z_{1}, x_{2}, x_{3}, L̄^{2}, z_{1}L̄)), and ||p_{2}^{I}||_{r} ≤ 4|H_{2}|′_{r} ≤ 4|e⃗|′_{r}.

p_{3}^{I} similarly has bounded norm and is convergent on Δ_{r}. These p^{I} functions are normalized as in (18). The other functions p_{1}, p_{4}, p_{5}, p_{a}^{R} will also, by construction, be of the form (14 − 17).

Equation (13), when using the normalized p⃗,

e_{5} = p_{5}^{E}(z_{1}, x_{2}, x_{3}, L̄^{2}) + z_{1}L̄p_{5}^{O}(z_{1}, x_{2}, x_{3}, L̄^{2})
−L̄(z_{1}^{2}p_{1}^{H}(L^{2}, x_{2}, x_{3}) + p_{1}^{A}(x_{2}, x_{3}, L̄^{2}))
−z_{1}(z̄_{1}^{2}\overline{p_{1}^{H}} + \overline{p_{1}^{A}} + p_{2}^{R} + ip_{3}^{R} + Re(p_{2}^{I}) + iRe(p_{3}^{I})),

can be solved next, by comparing coefficients of e_{5}(z_{1}, L̄, x_{2}, x_{3}) on the LHS:

e_{5} = ∑_{b=2r} e_{5}^{abcd}z_{1}^{a}L̄^{2r}x_{2}^{c}x_{3}^{d}+z_{1}L̄· ∑_{\begin{matrix}a > 0 \\ b = 2s + 1 \end{matrix}}e_{5}^{abcd}z_{1}^{a−1}L̄^{2s}x_{2}^{c}x_{3}^{d}+L̄ ∑_{b=2t+1} e_{5}^{0bcd}L̄^{2t}x_{2}^{c}x_{3}^{d},

to the RHS, where the only non-Q terms are those of L̄p_{1}^{A}. This decomposition of e_{5} determines the coefficients of the series for p_{1}^{A}(z_{2}, z_{3}, z_{4}) = ∑ p^{αβγ}z_{2}^{α}z_{3}^{β}z_{4}^{γ}, with p^{αβγ} = −e_{5}^{0,2γ+1,α,β}. The estimate follows:

3r||p_{1}^{A}||_{r} = 3r ∑ |p^{αβγ}|9^{γ}r^{α+β+2γ} = 3r ∑_{b=2t+1} | − e_{5}^{0bcd}|9^{t}r^{b−1+c+d}
= ∑ |e_{5}^{0bcd}|3^{b}r^{b+c+d} ≤ |e_{5}|′_{r} ≤ |e⃗|′_{r}
⟹ ||p_{1}^{A}||_{r} ≤ \frac{1}{3r} |e⃗|′_{r}.

An estimate for the conjugate \overline{p_{1}^{A}(x_{2}, x_{3}, L̄^{2})} will also be necessary. Formally expanding

\overline{p_{1}^{A}} = ∑ \overline{p^{αβγ}}L^{2γ}x_{2}^{α}x_{3}^{β} = ∑ q_{1}^{abc}z_{1}^{a}x_{2}^{b}x_{3}^{c}

gives, by Lemma 3.12, |q_{1}|′_{r} ≤ |p_{1}^{A}(x_{2}, x_{3}, L̄^{2})|′_{r} ≤ \frac{1}{3r} |e⃗|′_{r}.

Equation (12), again using the normalization, and expanding \overline{z_{1}^{2}p_{1}^{H}} as (L̄ − (x_{2} + ix_{3}))^{2}\overline{p_{1}^{H}}, becomes

e_{4} = p_{4}^{E}(z_{1}, x_{2}, x_{3}, L̄^{2}) + z_{1}L̄p_{4}^{O}(z_{1}, x_{2}, x_{3}, L̄^{2})

−2L̄((L̄^{2} + (x_{2} + ix_{3})^{2})\overline{p_{1}^{H}(L^{2}, x_{2}, x_{3})} + p_{2}^{R}(x_{2}, x_{3}) + ip_{3}^{R}(x_{2}, x_{3}))

+4L̄^{2}(x_{2} + ix_{3})\overline{p_{1}^{H}(L^{2}, x_{2}, x_{3})} − 2L̄\overline{p_{1}^{A}(x_{2}, x_{3}, L̄^{2})}
−2L̄(Re(p_{2}^{I}(z_{1}, x_{2}, x_{3}, L̄^{2}, z_{1}L̄)) + iRe(p_{3}^{I}(z_{1}, x_{2}, x_{3}, L̄^{2}, z_{1}L̄))).

This is handled by adding the known quantity 2L̄(\overline{p_{1}^{A}} + Re(p_{2}^{I}) + iRe(p_{3}^{I})) to both sides, and collecting the resulting LHS into

∑_{b=2r} k^{abcd}z_{1}^{a}L̄^{2r}x_{2}^{c}x_{3}^{d}+z_{1}L̄· ∑_{\begin{matrix}a ≥ 1 \\ b = 2s + 1 \end{matrix}} k^{abcd}z_{1}^{a−1}L̄^{2s}x_{2}^{c}x_{3}^{d}+L̄ ∑_{b=2t+1} k^{0bcd}L̄^{2t}x_{2}^{c}x_{3}^{d}.

Expressing LHS in terms of z_{1}, L̄, x_{2}, x_{3} means rearranging \overline{p_{1}^{A}(x_{2}, x_{3}, L̄^{2})} as a function of z_{1}, x_{2}, x_{3}, so that the previous estimate applies. Then p_{2}^{I} (and similarly p_{3}^{I}) can be expressed in terms of p′_{2}, whose norm was already bounded, and p′′_{2}, which is imaginary and does not contribute to Re(p_{2}^{I}):

e_{4}+2L̄(\overline{p_{1}^{A}}+Re(p_{2}^{I})+iRe(p_{3}^{I})) = e_{4}+2L̄(\overline{p_{1}^{A}}+ \frac{1}{2}(p′_{2}−2i ∑_{b odd} H_{2}^{0bcd}L̄^{b}x_{2}^{c}x_{3}^{d})

+\frac{i}{2}(p′_{3} − 2i ∑_{b odd} H_{3}^{0bcd}L̄^{b}x_{2}^{c}x_{3}^{d}))

|e_{4} + 2L̄(p̄_{1}^{A} + Re(p_{2}^{I}) + iRe(p_{3}^{I}))|′_{r} ≤ |e_{4}|′_{r} + 2 · 3r(\frac{1}{3r} |e⃗|′_{r} + 4|e⃗|′_{r})
≤ (3 + 24r)|e⃗|′_{r}.

Considering ∑_{b=2t+1} k^{0bcd}L̄^{2t}x_{2}^{c}x_{3}^{d} as a function of variables L̄^{2}, x_{2}, and x_{3},

by the Weierstrass Division Theorem there exist unique analytic functions 𝔮(L̄^{2}, x_{2}, x_{3}) and 𝔯(x_{2}, x_{3}) such that

∑_{b=2t+1} k^{0bcd}L̄^{2t}x_{2}^{c}x_{3}^{d} = (L̄^{2} + (x_{2} + ix_{3})^{2})𝔮(L̄^{2}, x_{2}, x_{3}) + 𝔯(x_{2}, x_{3}),

with 𝔮 = ∑ 𝔮^{rst}L̄^{2r}x_{2}^{s}x_{3}^{t} and 𝔯 = ∑ 𝔯^{st}x_{2}^{s}x_{3}^{t}. Let

p_{1}^{H}(z_{1}, z_{2}, z_{3}) = −\frac{1}{2} ∑ \overline{𝔮^{rst}}(z_{1} + z_{2} − iz_{3})^{2r}z_{2}^{s}z_{3}^{t},

and then the coefficients of p_{2}^{R} and p_{3}^{R} are the real and imaginary parts of the coefficients of −\frac{1}{2}𝔯(x_{2}, x_{3}). This is exactly where the second non-degeneracy condition is used; without both the x_{2}z̄_{1} and ix_{3}z̄_{1} terms in h_{4}, not all the higher-order terms could be eliminated.

The estimates from the Division Theorem are

|(x_{2} + ix_{3})^{2}|′_{r} = 4r^{2} < \frac{1}{2}|L̄^{2}|′_{r} = \frac{1}{2} · 9r^{2},
|L̄ ∑_{b=2t+1} k^{0bcd}L̄^{2t}x_{2}^{c}x_{3}^{d}|′_{r} ≤ (3 + 24r)|e⃗|′_{r},

|𝔮(L̄, x_{2}, x_{3})|′_{r} ≤ \frac{1}{9r^{2}} | ∑_{b=2t+1} k^{0bcd}L̄^{2t}x_{2}^{c}x_{3}^{d}|′_{r} · \frac{1}{1 − \frac{1}{2}} ≤ \frac{2}{9r^{2}} \frac{3 + 24r}{3r} |e⃗|′_{r},

|𝔯(x_{2}, x_{3})|′_{r} ≤ 2\frac{3 + 24r}{3r} |e⃗|′_{r}.

The remaining functions p_{4}^{E}, p_{4}^{O} can be estimated by solving (12):

||p_{4}||_{r} = |p_{4}^{E}(z_{1}, x_{2}, x_{3}, L̄^{2}) + z_{1}L̄p_{4}^{O}(z_{1}, x_{2}, x_{3}, L̄^{2})|′_{r}
= |e_{4} + 2L̄(\overline{p_{1}^{A}} + Re(p_{2}^{I}) + iRe(p_{3}^{I}))

+2L̄((L̄^{2} + (x_{2} + ix_{3})^{2})\overline{p_{1}^{H}} + p_{2}^{R} + ip_{3}^{R}) − 4L̄^{2}(x_{2} + ix_{3})\overline{p_{1}^{H}}|′_{r}

≤ ((3 + 24r) + (3 + 24r) + 4 · 9r^{2} · 2r · \frac{1}{9r^{2}} · \frac{3 + 24r}{3r})|e⃗|′_{r}
= (14 + 112r)|e⃗|′_{r}.

The estimates for the other components of p⃗ follow from Lemma 3.12 and the Banach algebra properties:

||p_{1}^{H}(z_{1}, z_{2}, z_{3})||_{r} ≤ \frac{1}{9r^{2}} \frac{3 + 24r}{3r} |e⃗|′_{r} = \frac{1 + 8r}{9r^{3}} |e⃗|′_{r},
||p_{1}||_{r} = ||z_{1}^{2}p_{1}^{H} + p_{1}^{A}||_{r} ≤ \frac{1 + 8r}{9r} |e⃗|′_{r} + \frac{1}{3r} |e⃗|′_{r} = (\frac{8}{9} + \frac{4}{9r})|e⃗|′_{r},
||p_{2}||_{r} = ||p_{2}^{I} + p_{2}^{R}||_{r} ≤ (12 + \frac{1}{r})|e⃗|′_{r},
||p_{3}||_{r} = ||p_{3}^{I} + p_{3}^{R}||_{r} ≤ (12 + \frac{1}{r})|e⃗|′_{r}.

Returning to Equation (13) to control p_{5},

||p_{5}||_{r} = |p_{5}^{E}(z_{1}, x_{2}, x_{3}, L̄^{2}) + z_{1}L̄p_{5}^{O}(z_{1}, x_{2}, x_{3}, L̄^{2})|′_{r}
= |e_{5}+L̄p_{1}−2z_{1}L̄(x_{2}+ix_{3})\overline{p_{1}^{H}}

+z_{1}(\overline{p_{1}^{A}}+Re(p_{2}^{I})+iRe(p_{3}^{I})+(L̄^{2}+(x_{2}+ix_{3})^{2})\overline{p_{1}^{H}} +p_{2}^{R}+ip_{3}^{R})|′_{r}

≤ (1 + 3r(\frac{8}{9} + \frac{4}{9r}) + 12r^{3}(\frac{1 + 8r}{9r^{3}} ) + r(\frac{1}{3r} + 4) + \frac{r}{3r} (3 + 24r))|e⃗|′_{r}
= (5 + \frac{76}{3}r)|e⃗|′_{r}.

Similar calculations, which appear in [C], solve the analogous linear problem for the m < n normal form.

3.7. The new defining equations. If the solution p⃗ to the linear problem, obtained in Theorem 3.16, is substituted into the nonlinear expressions (4 − 7), the resulting quadruple of formal series F (e⃗, p⃗) has its first two components with order at least 2d − 3, and the last two components with order at least 2d − 2. If p⃗ converges on a small polydisc Δ_{r}, then F (e⃗, p⃗) must be considered on even smaller polydisc, D_{ρ}, ρ < r, so that the composition of p⃗ and x_{2} + iH_{2}, etc., in Equations (4 − 7) is well-defined. Further, F (e⃗, p⃗) involves z_{1}, z̄_{1}, L and L̄, and even if these are rearranged to series in E or E′, the new defining equations are ready for another approximate coordinate change only after conversion to the z̃_{1}, x̃_{2}, x̃_{3} coordinates using the inverse transformation (2). Since there was also some flexibility in the normalization of the solution p⃗, the existence of a convergent solution of F (e⃗, p⃗) = 0_{E} remains an open question.

4. An explicit example of the solution

As a demonstration of the small radius of the the normalized solution of the linearized Equations (10 − 13), consider the following real algebraic variety in ℂ^{5}, with R > 0 and odd degree d ≥ 5:

M = \{z⃗ : y_{2} = y_{3} = 0, z_{4} = L̄^{2} + \frac{L̄^{d}}{R − z̄_{1}} , z_{5} = z_{1}L̄ + \frac{L̄x_{2}^{d−1}}{R − x_{2}} \}.

Clearly, for any ρ < R the variety is a graph over D_{ρ}, with e_{4} and e_{5} expressible as power series in A_{ρ}. The linearized functional equation for a coordinate change z⃗ + p⃗(z⃗) taking this variety to the normal form variety M̃^{4} becomes:

0 = Im(p_{2}(z_{1}, x_{2}, x_{3}, L̄^{2}, z_{1}L̄)) = Im(p_{3}(z_{1}, x_{2}, x_{3}, L̄^{2}, z_{1}L̄)),
\frac{L̄^{d}}{R − z̄_{1}} = p_{4}(z_{1}, x_{2}, x_{3}, L̄^{2}, z_{1}L̄) − 2L̄\overline{p_{1}(z_{1}, x_{2}, x_{3}, L̄^{2}, z_{1}L̄)}

−2L̄(Re(p_{2}(z_{1}, x_{2}, x_{3}, L̄^{2}, z_{1}L̄)) + iRe(p_{3}(z_{1}, x_{2}, x_{3}, L̄^{2}, z_{1}L̄))),

\frac{L̄x_{2}^{d−1}}{R − x_{2}} = p_{5}(z_{1}, x_{2}, x_{3}, L̄^{2}, z_{1}L̄)
−L̄p_{1}(z_{1}, x_{2}, x_{3}, L̄^{2}, z_{1}L̄) − z_{1}\overline{p_{1}(z_{1}, x_{2}, x_{3}, L̄^{2}, z_{1}L̄)}

−z_{1}(Re(p_{2}(z_{1}, x_{2}, x_{3}, L̄^{2}, z_{1}L̄)) + iRe(p_{3}(z_{1}, x_{2}, x_{3}, L̄^{2}, z_{1}L̄))).

In terms of the expressions (14 − 18), the normalized solution is:

p_{1}^{A} = \frac{−z_{2}^{d−1}}{R − z_{2}}

p_{1}^{H} = \frac{−(R + z_{2} − iz_{3})L^{2}}{2((R + z_{2} − iz_{3})^{2} − L^{2})} · ( ∑_{k=0}^{(d−5)/2}(−1)^{k}L^{d−5−2k}(z_{2} − iz_{3})^{2k})

−\frac{(−1)^{(d−3)/2}(R + z_{2} − iz_{3})^{3}(z_{2} − iz_{3})^{d−3}}{2((R + z_{2} − iz_{3})^{2} − L^{2})((R + z_{2} − iz_{3})^{2} + (z_{2} − iz_{3})^{2})}

p_{2}^{R} = \frac{x_{2}^{d−1}}{R − x_{2}} − \frac{1}{2}Re(\frac{(−1)^{(d−1)/2}(x_{2} + ix_{3})^{d−1}(R + x_{2} + ix_{3})}{(R + x_{2} + ix_{3})^{2} + (x_{2} + ix_{3})^{2}})
p_{3}^{R} = −\frac{1}{2}Im(\frac{(−1)^{(d−1)/2}(x_{2} + ix_{3})^{d−1}(R + x_{2} + ix_{3})}{(R + x_{2} + ix_{3})^{2} + (x_{2} + ix_{3})^{2}})

p_{4}^{E} = \frac{2(z_{2} + iz_{3})(R + z_{2} + iz_{3})z_{4}^{2}}{(R + z_{2} + iz_{3})^{2} − z_{4}} · ( ∑_{k=0}^{(d−5)/2}(−1)^{k}z_{4}^{(d−5)/2−k}(z_{2} + iz_{3})^{2k})

+\frac{(−1)^{(d−3)/2}2(R + z_{2} + iz_{3})^{3}(z_{2} + iz_{3})^{d−2}z_{4}}{((R + z_{2} + iz_{3})^{2} − z_{4})((R + z_{2} + iz_{3})^{2} + (z_{2} + iz_{3})^{2})}

+\frac{z_{4}^{(d+1)/2}}{(R + z_{2} + iz_{3})^{2} − z_{4}}

p_{5}^{E} = \frac{−z_{1}(R + z_{2} + iz_{3})z_{4}^{(d−1)/2}}{2((R + z_{2} + iz_{3})^{2} − z_{4})}

p_{5}^{O} = \frac{(z_{2} + iz_{3})(R + z_{2} + iz_{3})z_{4}}{(R + z_{2} + iz_{3})^{2} − z_{4}} · ( ∑_{k=0}^{(d−5)/2}(−1)^{k}z_{4}^{(d−5)/2−k}(z_{2} + iz_{3})^{2k})

+\frac{(−1)^{(d−3)/2}(R + z_{2} + iz_{3})^{3}(z_{2} + iz_{3})^{d−2}}{((R + z_{2} + iz_{3})^{2} − z_{4})((R + z_{2} + iz_{3})^{2} + (z_{2} + iz_{3})^{2})} + z_{1} · p_{1}^{H}.

The quantities p_{2}^{I}, p_{3}^{I}, and p_{4}^{O} are identically zero. All the p components should be functions of the z variables; in p_{2}^{R} and p_{3}^{R}, the x_{a} variables should be replaced by z_{a}, after calculating the real and imaginary parts. Also, the “L” appearing in p_{1}^{H} should be considered as L = z_{1}+z_{2}−iz_{3}. The functions are evidently holomorphic in a region around the origin of ℂ^{5}, outside the union of complex hypersurfaces where the denominators vanish. The linear factors in the denominator of p_{1}^{H} arising from the rearrangement are:

(R + z_{2} − iz_{3})^{2} − L^{2} = (R − z_{1})(R + z_{1} + 2(z_{2} − iz_{3})).

The factors in the denominators occurring from applying the Weierstrass Theorem to find 𝔮, 𝔯, and their conjugates, are:

(R+z_{2}+iz_{3})^{2}+(z_{2}+iz_{3})^{2} = (R+(1+i)z_{2}−(1−i)z_{3})(R+(1−i)z_{2}+(1+i)z_{3}),

(R+z_{2}−iz_{3})^{2}+(z_{2}−iz_{3})^{2} = (R+(1−i)z_{2}−(1+i)z_{3})(R+(1+i)z_{2}+(1−i)z_{3}). (The zero locus of the last two linear factors is the same pair of hyperplanes as defined by the first two.) Finally, the (R + z_{2} + iz_{3})^{2} − z_{4} appearing in the denominators of p_{4} and p_{5} is irreducible, and if z_{2} and z_{3} are bounded on the order of magnitude of R, then z_{4} must be bounded by some fraction of R^{2}. By inspection, all five components of p⃗ are convergent on Δ_{R/9} = 𝔻_{(R/9,R/9,R/9,R^{2}/9,R^{2}/27)}, as predicted by the estimates, and also on any other polydisc 𝔻_{r} not intersecting these complex hypersurfaces.

5. Analogy with singularity theory

If the equations given as a graph over the tangent space are considered as a real analytic map ℝ^{m} → ℝ^{2n}, they can be complexified by replacing each z̄_{s} by an independent holomorphic variable w_{s}. The graphing equations then become a holomorphic parametrization of a complex submanifold: ℂ^{m} → ℂ^{2n}. If the real manifold has complex tangents, then the composition of the embedding with the projection ℂ^{2n} → ℂ^{n} which forgets the new w variables is a singular map ℂ^{m} → ℂ^{n} (cf [W_{84}]). (To be more precise, these graphing equations and parametrizations should be considered as germs of mappings.)

Example 5.1. In the m = n = 2 case, Bishop ([Bishop]) normalized the defining equation to z_{2} = β(z_{1}^{2} + z̄_{1}^{2}) + z_{1}z̄_{1} + O(3) with β ≥ 0, where the parameter β is a holomorphic invariant, and describes the geometry of a complex tangent. The complexification and then projection of the quadratic terms is a map (z, w) ↦ (z, β(z^{2} + w^{2}) + zw). For β > 0, this is a ramified two-to-one map ([W_{84}]), and is analogous to Whitney’s “fold” singularity (z, w) ↦ (z, w^{2}).

Example 5.2. The β = 0 case, where a cubic normal form is z_{2} = z_{1}z̄_{1} + z̄_{1}^{3} ([M]), is similarly analogous to Whitney’s “cusp”, (x, y) ↦ (x, xy + y^{3}).

Example 5.3. For the normal form variety M̃^{4} in ℂ^{5}, a parametrization ℂ^{4} → ℂ^{10} → ℂ^{5} of the complexification H is

(z_{1}, w_{1}, z_{2}, z_{3}) ↦ (z_{1}, z_{2}, z_{3}, (w_{1} + z_{2} + iz_{3})^{2}, z_{1}(w_{1} + z_{2} + iz_{3})).

A normal form more like Whitney’s would be possible using a larger group, where the z and w variables could be transformed independently. Under the subgroup used to normalize the CR singularity, one expects equivalence classes of maps to be smaller, and continuous parameters (“moduli”) to appear “sooner” (for more and for lower-order terms). However, invariants which distinguish maps under the larger group will still distinguish them under the smaller group. Invariants of the complexification, such as the intrinsic derivative, the Boardman sequence, Jacobian extensions, etc., could provide a coarse but general beginning to a classification of CR singularities ([GG], [Porteous]).

Under the even smaller subgroup of transformations which preserve a given orientation of the tangent space at an isolated CR singularity, the + or − sign appearing in the quadratic normal form (Subsection 2.2) indicates a chirality near the complex tangent that is reversed by a change of orientation, and which corresponds to a notion of a topological intersection number, an “index” ±1. From a global viewpoint, isolated complex tangents of generic immersions of compact manifolds in ℂ^{n} can be enumerated by characteristic class formulas which relate the sum of indices to topological invariants. For example, when ℂP^{2} is smoothly embedded in ℂ^{5}, the expected sum of the indices of the complex tangents is p_{1}ℂP^{2} = 3. This is reminiscent of the appearance of cross-caps in maps from ℝP^{2} to ℝ^{3}, except that the real cross-caps can be eliminated in pairs to give a smooth immersion, and the CR cross-caps cannot be eliminated to give a totally real immersion. (For illustrations and explicit parametrizations of surfaces with real cross-cap singularities, see [Apéry], [CSS].) More generally, enumerative formulas hold for a real subbundle T of a complex vector bundle (F, J) over a real manifold M ([Wells], [L], [HL_{93}], [HL_{95}], [D_{1}], [D_{2}]). If the locus of complex tangents N_{j} = {x ∈ M : dim_{ℂ} T_{x} ∩ J_{x}T_{x} = j} forms a codimension 2j(n−m+j) real submanifold, N_{j+1} = Ø, and H_{x}^{j} = T_{x} ∩J_{x}T_{x} forms a bundle over N_{j}, then characteristic numbers of H^{j} → N_{j} are related to the pontrjagin classes of T and the chern classes of (F, J). Formulas of [W_{85}], [W_{86}] were shown in [C] to be special cases of degeneracy loci formulas of [Pragacz].

Formulas are also known which describe obstructions to the absence of complex tangent points with j > 1. There may also be topological obstructions for the non-degeneracy conditions (Subsection 2.2), so that degenerate points occur as a subset of the locus of complex tangents N_{1} ⊆ M, generically of lower dimension. The assumptions on the rank, made in arriving at the normal form, represent the most common and most important case; both non-degeneracy conditions were used in the formal stability Theorem, but there may be similar stability results in some degenerate cases.

Another phenomenon related to complex tangents generalizes the “coincidence of complex structures” of [HL_{95}]. A smooth map f : M_{1} → M_{2} between complex manifolds (M_{1}, J_{1}), (M_{2}, J_{2}), may respect the complex structures, that is, df(J_{1}(v_{x})) = J_{2}(df(v_{x})), for some tangent vectors v_{x} at some points x ∈ M_{1}. The generic behavior of this construction, which more generally applies to real linear maps between complex vector bundles, can be described by enumerative formulas in chern classes; see [EW], [HL_{95}], [C]. The graph of a smooth map f inside M_{1} ×M_{2} (or the graph of a bundle map inside the direct sum) has complex tangents exactly at points of coincidence of the complex structures. For example, the differential of a smooth map from a complex surface to a complex 3-manifold generically will commute with the complex structure only at isolated points x, and only for a complex line of vectors in T_{x}. The graph is a real 4-manifold with isolated complex tangents inside a product space of five complex dimensions, and again the cross-cap singularity of real surfaces in 3-space comes to mind.

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Received Received date / Revised version date

Department of Mathematical Sciences Indiana University-Purdue University Fort Wayne Fort Wayne, IN 46805-1499

E-mail address: CoffmanA@ipfw.edu
http://www.ipfw.edu/math/Coffman/