Analytic Normal Form for CR Singular Surfaces in ℂ^{3}

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

                    CoffmanA@ipfw.edu

August 23, 2004

Abstract

A real analytic surface inside complex 3-space with an isolated, nondegenerate complex tangent is shown to be biholomorphically equivalent to a fixed real algebraic variety. The analyticity of the normalizing transformation is proved using a rapid convergence argument. Real surfaces in higher dimensions are also shown to have an algebraic normal form. ^{1} ^{2}

1 Introduction

This paper contributes to the program of studying the “local equivalence problem for real submanifolds of ℂ^{n},” as described in the survey paper, [BER]. The main result (Theorem 5.7) establishes a normal form for real analytic 2-manifolds in ℂ^{3}, near a suitably non-degenerate complex tangent, under biholomorphic coordinate changes. The non-degeneracy condition is an easily checked property of the quadratic terms of the equations that define the submanifold. The normal form is a fixed real algebraic variety, so there are no biholomorphic invariants for real surfaces near a non-degenerate complex tangent in ℂ^{3}. We will also point out some analogies between this normal form and some of Whitney’s equations from the singularity theory of smooth maps from surfaces to ℝ^{3}.

1MSC 2000 32V40, 32S05

2Key words: normal form, CR singularity, complex tangent, rapid convergence

To outline the plan for the proof of the main result, let M be a real analytic surface in ℂ^{3}, which has an “isolated CR singularity” at the point x⃗ ∈ M, that is, the tangent plane at x⃗ is a complex line, but the tangent plane at all other points in a neighborhood of x⃗ is totally real. By a complex affine coordinate change, it can be assumed that the singularity is at the origin, and that the tangent plane is one of the three complex coordinate axes. The local geometry of M, in some polydisc Δ centered at the singularity, will be considered first by specifying a quadratic non-degeneracy condition (the inequality (1) in Section 2), and in the case where M satisfies this condition, setting up a system of functional equations whose solution, if it exists, is a “normalizing transformation,” meaning a coordinate change so that in the new coordinate system, the defining equations for M are in the real algebraic normal form (equation (4) in Section 2). Since an exact solution to these non-linear equations (equations (8) and (9) in Section 3) is hard to find in just one step, an approximation p⃗ to this solution will be found by solving a related system of linear equations ((10) and (11)). The solution technique for the linear equations is a straightforward comparison of coefficients, and p⃗ and its derivatives will turn out to be bounded in a neighborhood comparable in size to Δ. The size of the domain of p⃗ is a crucial ingredient of the analysis, so that when we get a sequence of such approximate solutions, each of them will be analytic on a certain fixed polydisc. The rapid convergence technique employed in Section 5 closely follows the construction of [M], where a convergent normalizing transformation was proved to exist for a surface in ℂ^{2} with a special kind of CR singularity.

In the final Section, a cubic normal form is derived for surfaces in ℂ^{n}, n > 3. Again, being real analytic and suitably non-degenerate will be enough to guarantee the existence of a coordinate system on an open set, in which the surface is real algebraic.

This work benefited from the author’s conversations with Xianghong Gong.

2 The non-degeneracy condition, and a quadratic normal form

Let (z_{1}, z_{2}, z_{3}) be a coordinate system for ℂ^{3}, and assume M is a real analytic surface in ℂ^{3} with a complex tangent at 0 ∈ M. By a complex linear transformation, the tangent plane T0M can be assumed to be the z_{1}-axis. (z_{1} will hereafter be abbreviated as z.) Then there is some polydisc Δ centered at 0 so that the defining equations of M in Δ are in the form of a graph over a disc in the z-axis:

z_{2} = h_{2}(z, z̄),
z_{3} = h_{3}(z, z̄),

where h_{2} and h_{3} are complex-valued real analytic functions defined in a neighborhood of z = 0, and vanishing to second order at z = 0. These functions can be expressed as the restriction to {ζ = z̄} of the two-variable series:

h_{2}(z, ζ) = α_{2}z^{2} + β_{2}zζ + γ_{2}ζ^{2} + ∑_{a+b≥3} h_{2}^{a,b}z^{a}ζ^{b},
h_{3}(z, ζ) = α_{3}z^{2} + β_{3}zζ + γ_{3}ζ^{2} + ∑_{a+b≥3} h_{3}^{a,b}z^{a}ζ^{b},

each of which converges on the set {(z, ζ) : |z| < R, |ζ| < R} to a complex analytic function. The terms α_{2}z^{2} and α_{3}z^{2} can be eliminated by a holomorphic quadratic coordinate change (which may change the above radius R). From this point, M will be assumed to satisfy the non-degeneracy condition:

det ( \begin{matrix}β_{2} & γ_{2} \\ β_{3} & γ_{3} \end{matrix} ) ≠ 0,  (1)

so that there is a complex linear transformation of z_{2} and z_{3} bringing M to the following quadratic normal form:

z_{2} = z̄^{2} + e_{2}(z, z̄),  (2)
z_{3} = zz̄ + e_{3}(z, z̄),  (3)

with e_{2} and e_{3} complex-valued real analytic functions vanishing to third order at 0.

Let M̃ denote the real algebraic surface where e_{2} and e_{3} are identically zero:

M̃ = {(z, z_{2}, z_{3}) ∈ ℂ^{3} : z_{2} = z̄^{2}, z_{3} = zz̄}.  (4)

The main result (Theorem 5.7) is that if M satisfies the non-degeneracy condition, and is described by equations of the form (2), (3), then there exist holomorphic functions p_{1}(z, z_{2}, z_{3}), p_{2}(z, z_{2}, z_{3}), p_{3}(z, z_{2}, z_{3}) defined on some polydisc centered at 0 so that in the following new local coordinate system for ℂ^{3}:

z̃ = z + p_{1}  (5)
z̃_{2} = z_{2} + p_{2}  (6)
z̃_{3} = z_{3} + p_{3},  (7)

M is defined, in a neighborhood of 0, by z̃_{2} = ¯ z̃^{2}, z̃_{3} = z¯z̃; that is, M and M̃ are locally analytically equivalent.

It is worth mentioning that M̃ is totally real except at the origin, and that it is contained in the singular complex hypersurface z^{2}z_{2} − z_{3}^{2} = 0. The equations for M̃ apparently resemble Whitney’s parametric equations for a surface with a cross-cap singularity in ℝ^{3} ([W]),

(u, v) ↦ (x, y, z) = (u, v^{2}, uv),

and the equation for the complex hypersurface resembles the implicit equation x^{2}y − z^{2} = 0 for the Whitney umbrella variety in ℝ^{3}.

Before starting the proof of the main result, we conclude this Section by comparing the situation of surfaces in ℂ^{3} with the situation in higher dimensions.

In [C_{1}], the local equivalence problem was considered for m-submanifolds of ℂ^{n}, with \frac{2}{3}(n + 1) ≤ m < n (note that the case m = 2, n = 3 is outside this range). The approach of [C_{1}] was, as in this paper, to first establish non-degeneracy conditions (analogous to (1)) for the quadratic part of the defining equations, and then to show there is a polynomial normal form, such that near a non-degenerate complex tangent of a real analytic m-submanifold, there exists a formal coordinate change so that the new defining equations are in the polynomial normal form. For example, in the m = 4, n = 5 case, the normal form variety in the 5-space with coordinates (z_{1}, z_{2}, z_{3}, z_{4}, z_{5}), z_{a} = x_{a} + iy_{a}, is a graph over the real subspace with coordinates z_{1}, x_{2}, x_{3}:

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

The obvious difference between these equations and those defining the surface M̃ is that the normal form for the surface can be expressed using only monomials. This is significant, because the analysis of 4-manifolds in [C_{1}] was complicated by the presence of multinomial coefficients appearing in powers of the quantity (z̄_{1} + x_{2} + ix_{3}).

In fact, the approximate normalizing transformations for the local equivalence problem in [C_{1}] were not suitable for the “rapid convergence” technique. While a formal power series for the normalizing transformation could be constructed as a formal composition of a sequence of holomorphic approximations, each of these approximations was defined on a polydisc significantly smaller than the previous one. So, the intersection of the domains of the transformations was just the single point at the origin. The question of convergence or divergence of the formal coordinate change for higher dimensions remains open. In the case of surfaces considered in this article, the rate of decrease of the size of the domain of the approximations is so small that we are able to apply the rapid convergence technique, to show that the approximations do indeed converge to an analytic coordinate change on an open set.

Some of the calculations of the next Section will be similar to those in [C_{1}], but simpler and with fewer variables. The goal is, given real analytic functions e_{2} and e_{3} (as in equations (2), (3)), to construct complex analytic functions p_{1}, p_{2}, p_{3} (as in (5), (6), (7)). This goal will not be met until Section 5, but along the way we will state each major step as a “Theorem.” The first Theorem decomposes e_{2} and e_{3} into even and odd parts, for the purposes of a comparison of coefficients, to arrive at an approximate normalizing transformation as the solution of a linearized functional equation. The even/odd decomposition also has some similarities with the argument of [W] proving the stability of the cross-cap singularity.

It should also be remarked that in the dimension range \frac{2}{3}(n + 1) ≤ m < n, CR singularities of m-submanifolds of ℂ^{n} are “topologically stable,” roughly meaning that if a submanifold in sufficiently general position has a point where the tangent space contains a complex line, then small smooth perturbations of the manifold will also have this property. (The notion of general position can be defined in terms of intersection properties of the gauss map from the manifold to the grassmannian of real subspaces in ℂ^{n} — see [C_{1}], [G].) Surfaces in ℂ^{3} do not enjoy this topological stability property: a CR singular surface can always be perturbed to a nearby surface which is totally real at every point.

3 A functional equation

The following notation will be convenient:

Notation 3.1. For r = (r_{1}, …, r_{n}) ∈ ℝ^{n}, with all r_{k} > 0, define a polydisc (with center 0) in ℂ^{n} by

𝔻_{r} = {(z_{1}, …, z_{n}) : |z_{k}| < r_{k}}.

As special cases, let

D_{r} = 𝔻_{(r,r)}

and

Δ_{r} = 𝔻_{(r,r^{2},r^{2})}.

Often in this paper, we will discuss both multivariable power series with center 0 and complex coefficients, which converge on a polydisc 𝔻_{r}, and complex valued functions which are complex analytic on a polydisc 𝔻_{r}. It is well-known that these two notions are equivalent, so we will not always explicitly mention that. It is also well-known that a function which is real analytic on a “real polydisc,” 𝔻_{r} ∩ ℝ^{n}, may have a series expansion with center 0 that does not converge on all of 𝔻_{r} ∩ ℝ^{n}, so when working with real analytic functions, we will be careful about the domain of convergence, mostly by assuming that the real polydisc is small enough so that the function is the restriction of a complex analytic function on a complex polydisc.

Notation 3.2. For a complex-valued function e(z, ζ) of two complex variables, which is defined on some set containing the polydisc D_{r}, define the norm

|e|_{r} = sup_{(z,ζ)∈D_{r}} |e(z, ζ)|.

For a pair e⃗ = (e_{2}, e_{3}), define

|e⃗|_{r} = max{|e_{2}|_{r}, |e_{3}|_{r}}.

For a complex-valued function p(z, z_{2}, z_{3}) of three complex variables, which is defined on some set containing the polydisc Δ_{r}, define the norm

∥p∥_{r} = sup_{(z,z_{2},z_{3})∈Δ_{r}} |p(z, z_{2}, z_{3})|.

Notation 3.3. For a series in two variables e(z, ζ) = ∑ e^{a,b}z^{a}ζ^{b}, define its “degree” as the lowest integer a + b so that e^{a,b} ≠ 0. (Also, the “degree” of a pair of series is the lower of the two degrees.) For a series p(z, z_{2}, z_{3}) = ∑ p^{abc}z^{a}z_{2}^{b}z_{3}^{c}, define its “weight” as the lowest integer a + 2b + 2c so that p^{abc} ≠ 0.

With this notation, and a real analytic manifold M defined by equations (2), (3), we can assume that there is some r > 0 so that |(e_{2}(z, ζ), e_{3}(z, ζ))|_{r} is finite. Given e⃗ = (e_{2}, e_{3}) with degree ≥ 3, the goal is to find some r̃, 0 < r̃ ≤ r, and some holomorphic functions p⃗ = (p_{1}, p_{2}, p_{3}) on Δr̃, so that the transformation

Ψ(z, z_{2}, z_{3}) = (z̃, z̃_{2}, z̃_{3})

defined by (5), (6), (7) is a biholomorphism with domain Δr̃ taking M to M̃. That is, if (z, z_{2}, z_{3}) ∈ M ∩ Δr̃, then z̃_{2} = ¯ z̃^{2} and z̃_{3} = z̃¯z̃. The weight of p_{1} will be ≥ 2 and the weight of p_{2} and p_{3} will be ≥ 3.

For now, suppose Ψ is any (not necessarily invertible) map of the form (5), (6), (7). The following equations must hold for any (z, z_{2}, z_{3}) ∈ M close enough to 0 so that both LHS and RHS are defined:

z̃_{2} − ¯ z̃^{2} = z_{2} + p_{2}(z, z_{2}, z_{3}) − \overline{(z + p_{1}(z, z_{2}, z_{3}))}^{2}
= e_{2}(z, z̄) + p_{2}(z, z̄^{2} + e_{2}, zz̄ + e_{3})  (8)
−(2z̄\overline{p_{1}(z, z̄^{2} + e_{2}, zz̄ + e_{3})} + \overline{p_{1}(z, z̄^{2} + e_{2}, zz̄ + e_{3})}^{2}),
z̃_{3} − z̃¯z̃ = z_{3} + p_{3}(z, z_{2}, z_{3}) − (z + p_{1}(z, z_{2}, z_{3}))\overline{(z + p_{1}(z, z_{2}, z_{3}))}
= e_{3}(z, z̄) + p_{3}(z, z̄^{2} + e_{2}, zz̄ + e_{3})  (9)
−(z\overline{p_{1}(z, z̄^{2} + e_{2}, zz̄ + e_{3})} + z̄p_{1}(z, z̄^{2} + e_{2}, zz̄ + e_{3}))
−|p_{1}(z, z̄^{2} + e_{2}, zz̄ + e_{3})|^{2}.

If there are holomorphic functions p⃗ on some Δr̃ so that Ψ is a biholomorphism, and the quantities (8), (9) are identically 0 (as formal series in z, z̄), then Ψ is a normalizing transformation. As a first step in solving for p⃗ in terms of e⃗, consider the following two simpler equations:

0 = e_{2}(z, z̄) + p_{2}(z, z̄^{2}, zz̄) − 2z̄\overline{p_{1}(z, z̄^{2}, zz̄)},  (10)
0 = e_{3}(z, z̄) + p_{3}(z, z̄^{2}, zz̄) − z\overline{p_{1}(z, z̄^{2}, zz̄)} − z̄p_{1}(z, z̄^{2}, zz̄). (11)

To see how the new equations are related to the original system, suppose e⃗ has degree d ≥ 3, and that p⃗ = (p_{1}, p_{2}, p_{3}) is a solution of (10), (11) so that p_{1} has weight ≥ d − 1, and p_{2} and p_{3} have weight ≥ d. Using this solution p⃗ to evaluate the RHS of (8) and (9) evidently results in expressions of degree ≥ 2d − 2. So, Ψ defined by p⃗ may not bring M to normal form, but it will approximately double the degree of the higher-order terms.

Theorem 3.4. Given r > 0 and a pair of complex analytic functions e⃗ = (e_{2}(z, ζ), e_{3}(z, ζ)) defined on D_{r} with |e⃗|_{r} < ∞ and degree d ≥ 3, there exists a triple of power series p⃗ = (p_{1}, p_{2}, p_{3}) which is complex analytic on Δ_{r}, and which exactly solves equations (10), (11). This solution satisfies the estimates

∥p_{1}∥_{r} ≤ \frac{3}{2r} |e⃗|_{r}, ∥p_{2}∥_{r} ≤ 5|e⃗|_{r}, ∥p_{3}∥_{r} ≤ 5|e⃗|_{r}.

Proof. First, notice that if p⃗(z, z_{2}, z_{3}) is a formal series solution of (10), (11), it certainly does not follow that p⃗ is convergent. For example, p_{2}(z, z_{2}, z_{3}) and p_{2}+(z^{2}z_{2}−z_{3}^{2})·Q are formally the same when restricted to {z_{2} = z̄^{2}, z_{3} = zz̄}, for any (possibly divergent) series Q(z, z_{2}, z_{3}). So, if one formal solution p⃗ exists, there exist infinitely many divergent solutions. The following choice of normalization will not only determine p⃗ uniquely, it will also result in the claimed convergence and bounds.

p_{1}(z, z_{2}, z_{3}) = p_{1}^{H}(z) + p_{1}^{A}(z_{2})
p_{2}(z, z_{2}, z_{3}) = p_{2}^{E}(z, z_{2}) + z_{3}p_{2}^{O}(z, z_{2})
p_{3}(z, z_{2}, z_{3}) = p_{3}^{E}(z, z_{2}) + z_{3}p_{3}^{O}(z, z_{2})

Also, p_{1}^{H} must be an even function of z: p_{1}^{H}(z) = p_{1}^{H}(−z), and the weights of p_{1}, p_{2}, p_{3} must be ≥ d − 1, d, d, respectively.

This choice of normalization can be motivated by inspecting the set of formal transformations which leave M̃ invariant; the normalizing transformations for M should in some sense be complementary to this set. This topic will be deferred to Section 6; for now, notice only that any formal series P (z, z_{2}, z_{3}) can, by the Weierstrass Division Theorem, be uniquely expanded in the form

P = (z^{2}z_{2} − z_{3}^{2}) · Q + z_{3}P^{O}(z, z_{2}) + P^{E}(z, z_{2}).

The last two “remainder” terms are as in the normalization of p_{2} and p_{3}. Also, any monomial e^{a,b}z^{a}z̄^{b} fits into one of the three cases:

• e^{a,2k}z^{a}(z̄^{2})^{k} = P^{E}(z, z̄^{2}), for b even, or

• (zz̄)e^{a,2k+1}z^{a−1}(z̄^{2})^{k} = zz̄P^{O}(z, z̄^{2}), for b odd and a > 0, or

• e^{0,2k+1}z̄^{2k+1}, with a = 0, b odd.

In the last case, the monomial is not equal to any expression of the form P (z, z̄^{2}, zz̄). So, the main idea of this proof will be to construct a function p_{1} which simultaneously solves the e^{0,b}z̄^{b}, b odd, components of equations (10), (11).

Combining equation (10) with the normalization conditions gives:

∑ e_{2}^{a,b}z^{a}z̄^{b} = 2z̄\overline{(p_{1}^{H}(z) + p_{1}^{A}(z̄^{2}))} − p_{2}(z, z̄^{2}, zz̄).

Since 2z̄\overline{p_{1}^{H}} is the only term on the RHS that could have monomials of the form z^{0}z̄^{b}, with b odd, p_{1}^{H} is determined uniquely and is even in z:

2z̄\overline{p_{1}^{H}(z)} = ∑_{b=2k+1} e_{2}^{0,b}z̄^{b}
2zp_{1}^{H}(z) = \overline{∑_{b=2k+1} e_{2}^{0,b}z̄^{b}}

p_{1}^{H}(z) = \frac{1}{4z} \overline{(e_{2}(0, z̄) − e_{2}(0, −z̄))}.

Of course, for |z| < r, the expression e_{2}(0, z̄) refers to the complex analytic function e_{2}(z, ζ), evaluated at (0, z̄) ∈ D_{r}; it doesn’t mean z = z̄ = 0. Also, the division by z in the above expression for p_{1}^{H} should be taken formally, by subtraction of exponents in the formal power series, so that after the division, the series still has only positive exponents and converges to a complex analytic function on a domain containing z = 0. These conventions will be observed in some constructions that follow.

It follows from the Schwarz Lemma ([A]) that for |z| < r,

|\overline{e_{2}(0, z̄)}| ≤ \frac{|z|}{r} sup_{|z|<r} |\overline{e_{2}(0, z̄)}| ≤ \frac{|z|}{r}|e_{2}|_{r},

and,

∥p_{1}^{H}(z)∥_{r} = sup_{|z|<r} |\frac{\overline{(e_{2}(0, z̄) − e_{2}(0, −z̄))}}{4z}| ≤ \frac{1}{2r} |e_{2}|_{r}.

A similar substitution into equation (11):

∑ e_{3}^{a,b}z^{a}z̄^{b} = z̄(p_{1}^{H}(z) + p_{1}^{A}(z̄^{2})) + z\overline{p_{1}(z, z̄^{2})} − p_{3}(z, z̄^{2}, zz̄),

determines p_{1}^{A}(z_{2}) uniquely:

z̄p_{1}^{A}(z̄^{2}) = ∑_{b=2k+1} e_{3}^{0,b}z̄^{b}

ζp_{1}^{A}(ζ^{2}) = \frac{1}{2}(e_{3}(0, ζ) − e_{3}(0, −ζ))
p_{1}^{A}(z_{2}) = ∑_{k} e_{3}^{0,2k+1}z_{2}^{k}.

Again, the Schwarz Lemma gives a bound for the values:

∥p_{1}^{A}(z_{2})∥_{r} = sup_{|ζ|<r} |p_{1}^{A}(ζ^{2})| = sup_{|ζ|<r} |\frac{e_{3}(0, ζ) − e_{3}(0, −ζ)}{2ζ}|

≤ sup_{|ζ|<r} \frac{2\frac{|ζ|}{r}|e_{3}|_{r}}{|2ζ|} = \frac{1}{r}|e_{3}|_{r}.

It is now possible to solve equation (10) for p_{2}:

p_{2}^{E}(z, z̄^{2}) + zz̄p_{2}^{O}(z, z̄^{2}) = −e_{2}(z, z̄) + 2z̄\overline{(p_{1}^{H}(z) + p_{1}^{A}(z̄^{2}))}
= −e_{2}(z, z̄) + ∑_{b=2k+1} e_{2}^{0,b}z̄^{b} + 2\frac{z̄}{z}\overline{∑_{b=2k+1} e_{3}^{0,b}z̄^{b}}.

Replacing z̄ with ζ and separating both RHS and LHS into the even and odd parts with respect to ζ determines p_{2}^{E} and p_{2}^{O}:

p_{2}^{E}(z, ζ^{2}) = −\frac{1}{2}(e_{2}(z, ζ) + e_{2}(z, −ζ)),
p_{2}^{E}(z, z_{2}) = − ∑_{b=2k} e_{2}^{a,b}z^{a}z_{2}^{k},
∥p_{2}^{E}∥_{r} ≤ |e_{2}|_{r}.
zζp_{2}^{O}(z, ζ^{2}) = −\frac{1}{2}(e_{2}(z, ζ) − e_{2}(z, −ζ)) + \frac{1}{2}(e_{2}(0, ζ) − e_{2}(0, −ζ))

+2\frac{ζ}{z} \overline{\frac{1}{2}(e_{3}(0, z̄) − e_{3}(0, −z̄))},

p_{2}^{O}(z, ζ^{2}) = \frac{e_{2}(z, −ζ) − e_{2}(z, ζ) + e_{2}(0, ζ) − e_{2}(0, −ζ)}{2zζ}

+\frac{\overline{e_{3}(0, z̄) − e_{3}(0, −z̄)}}{z^{2}},

p_{2}^{O}(z, z_{2}) = ∑_{a≥1,b=2k+1} e_{2}^{a,b}z^{a−1}z_{2}^{k} + ∑_{b=2k+1} \overline{e_{3}^{0,b}}z^{2k−1},

∥p_{2}^{O}∥_{r} ≤ sup_{|z|<r} sup_{|ζ|<r} |\frac{e_{2}(z, −ζ) − e_{2}(z, ζ) + e_{2}(0, ζ) − e_{2}(0, −ζ)}{2zζ}|

+ sup_{|z|<r} |\frac{\overline{e_{3}(0, z̄) − e_{3}(0, −z̄)}}{z^{2}} |

≤ \frac{2}{r^{2}} |e_{2}|_{r} + \frac{2}{r^{2}} |e_{3}|_{r},

the last inequality following from several applications of the Schwarz Lemma. The calculation and bounds for p_{3} follow from a similar treatment of equation (11).

p_{3}^{E}(z, z̄^{2}) + zz̄p_{3}^{O}(z, z̄^{2}) = −e_{3}(z, z̄) + z̄p_{1}(z, z̄^{2}) + z\overline{p_{1}(z, z̄^{2})}
= −e_{3}(z, z̄) + \frac{z̄}{2z} \overline{∑_{b=2k+1} e_{2}^{0,b}z̄^{b}} + ∑_{b=2k+1} e_{3}^{0,b}z̄^{b}

+\frac{z}{2z̄} ∑_{b=2k+1} e_{2}^{0,b}z̄^{b} + \overline{∑_{b=2k+1} e_{3}^{0,b}z̄^{b}}.

p_{3}^{E}(z, ζ^{2}) = −\frac{1}{2}(e_{3}(z, ζ) + e_{3}(z, −ζ)) + \frac{z}{4ζ} (e_{2}(0, ζ) − e_{2}(0, −ζ))

+\frac{1}{2}\overline{(e_{3}(0, z̄) − e_{3}(0, −z̄))},

∥p_{3}^{E}∥_{r} ≤ 2|e_{3}|_{r} + \frac{1}{2}|e_{2}|_{r}.
zζp_{3}^{O}(z, ζ^{2}) = −\frac{1}{2}(e_{3}(z, ζ) − e_{3}(z, −ζ)) + \frac{1}{2}(e_{3}(0, ζ) − e_{3}(0, −ζ))

+\frac{ζ}{4z} \overline{(e_{2}(0, z̄) − e_{2}(0, −z̄))},

p_{3}^{O}(z, ζ^{2}) = \frac{e_{3}(z, −ζ) − e_{3}(z, ζ) + e_{3}(0, ζ) − e_{3}(0, −ζ)}{2zζ}

+\frac{\overline{e_{2}(0, z̄) − e_{2}(0, −z̄)}}{4z^{2}},

∥p_{3}^{O}∥_{r} ≤ \frac{2}{r^{2}} |e_{3}|_{r} + \frac{1}{2r^{2}} |e_{2}|_{r}.

It is not yet claimed that p⃗ defines a biholomorphism Ψ; this will be shown later (Theorem 4.4), under certain conditions on e⃗ and r. In Section 7, the above procedure for finding p⃗ will be demonstrated for a specific choice of e⃗.

Corollary 3.5. There is a constant c_{1} > 0 such that for any p⃗ and e⃗ as in Theorem 3.4, and any radii ρ, r with \frac{1}{2} < ρ < r ≤ 1, the following estimates hold:

max_{j=1,2,3}{∥p_{j}∥_{r}} ≤ c_{1}|e⃗|_{r},

max_{j=1,2,3} \{∑_{i=1}^{3} ∥\frac{dp_{i}}{dz_{j}} ∥_{ρ}\} ≤ \frac{c_{1}|e⃗|_{r}}{r − ρ} .

Proof. The bound ∥p_{1}∥_{r} < 3|e⃗|_{r} follows from the estimate from the previous Theorem and r > \frac{1}{2}. The bounds for the derivatives of p_{i} follow from an estimate which is a consequence of Cauchy’s estimate ([A]):

If 0 < R_{2} < R_{1} and f(z) is holomorphic and bounded by M for |z| < R_{1}, then \frac{df}{dz} is bounded by \frac{M}{R_{1}−R_{2}} for |z| < R_{2}.

This fact can be applied immediately to the bound ∥p_{j}∥_{r} ≤ 5|e⃗|_{r}, with R_{1} − R_{2} = r − ρ for the z derivatives, and R_{1} − R_{2} = r^{2} − ρ^{2} > r − ρ for the z_{2}, z_{3} derivatives. By inspecting the derivatives of the normalized functions, one could find the constant c_{1}, or improve upon the bounds claimed by the Theorem. Some estimates are recorded below, for use in the proof of Theorem 4.2. Note that not all the terms have r − ρ in the denominator, but \frac{1}{r−ρ} > 2, and it will be simpler in some of the later Theorems to treat all the quantities in the same way instead of using the sharpest estimates available.

∥\frac{dp_{1}}{dz} ∥_{ρ} ≤ \frac{\frac{1}{2r} |e_{2}|_{r}}{r − ρ}

∥\frac{dp_{1}}{dz_{2}} ∥_{ρ} ≤ \frac{\frac{1}{r}|e_{3}|_{r}}{r^{2} − ρ^{2}}
\frac{dp_{1}}{dz_{3}} = 0,
∥\frac{dp_{2}}{dz} ∥_{ρ} ≤ ∥\frac{dp_{2}^{E}}{dz} ∥_{ρ} + ∥z_{3} \frac{dp_{2}^{O}}{dz} ∥_{ρ} ≤ \frac{|e_{2}|_{r}}{r − ρ} + \frac{ρ^{2}}{r^{2}} \frac{2(|e_{2}|_{r} + |e_{3}|_{r})}{r − ρ}
∥\frac{dp_{2}}{dz_{2}} ∥_{ρ} ≤ ∥\frac{dp_{2}^{E}}{dz_{2}} ∥_{ρ} + ∥z_{3} \frac{dp_{2}^{O}}{dz_{2}} ∥_{ρ} ≤ \frac{|e_{2}|_{r}}{r^{2} − ρ^{2}} + \frac{ρ^{2}}{r^{2}} \frac{2(|e_{2}|_{r} + |e_{3}|_{r})}{r^{2} − ρ^{2}}
∥\frac{dp_{2}}{dz_{3}} ∥_{r} = ∥p_{2}^{O}∥_{r} ≤ \frac{2}{r^{2}} (|e_{2}|_{r} + |e_{3}|_{r})
∥\frac{dp_{3}}{dz} ∥_{ρ} ≤ ∥\frac{dp_{3}^{E}}{dz} ∥_{ρ} + ∥z_{3} \frac{dp_{3}^{O}}{dz} ∥_{ρ} ≤ (1 + \frac{ρ^{2}}{r^{2}} ) \frac{2|e_{3}|_{r} + \frac{1}{2}|e_{2}|_{r}}{r − ρ}

∥\frac{dp_{3}}{dz_{2}} ∥_{ρ} ≤ ∥\frac{dp_{3}^{E}}{dz_{2}} ∥_{ρ} + ∥z_{3} \frac{dp_{3}^{O}}{dz_{2}} ∥_{ρ} ≤ (1 + \frac{ρ^{2}}{r^{2}} ) \frac{2|e_{3}|_{r} + \frac{1}{2}|e_{2}|_{r}}{r^{2} − ρ^{2}}
∥\frac{dp_{3}}{dz_{3}} ∥_{r} = ∥p_{3}^{O}∥_{r} ≤ \frac{2}{r^{2}} |e_{3}|_{r} + \frac{1}{2r^{2}} |e_{2}|_{r}.

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 (z, z_{2}, z_{3}) ↦ (sz, s^{2}z_{2}, s^{2}z_{3}), s > 0, the equations (2), (3) can be assumed to define M for |z| < 1, and for any η > 0, there is a rescaling making |e⃗|_{1} < η. Such scalings form a subgroup of the stabilizer of the normal form, considered in Section 6.

4 The new defining equations and some estimates

The following Lemma on the ℓ_{1} norm in ℂ^{n} will be used several times:

Lemma 4.1. Let f(z⃗) = (f_{1}(z_{1}, …, z_{n}), …, f_{n}(z_{1}, …, z_{n})) be a holomorphic map f : 𝔻_{r} → ℂ^{n}, with

max_{j=1,...,n} \{∑_{i=1}^{n} sup_{z⃗∈𝔻_{r}} |\frac{df_{i}}{dz_{j}} (z⃗)|\} ≤ K.

Then, for z⃗, z ′ ∈ 𝔻_{r},

∑_{i=1}^{n} |f_{i}(z ′) − f_{i}(z⃗)| ≤ K ∑_{i=1}^{n} |z′_{i} − z_{i}|.

Proof. The single variable case, with n = 1 and f : 𝔻_{(r_{1})} → ℂ, follows from integrating \frac{df}{dz_{1}} along a path γ which is contained in the disc and connects z to z′:

|f(z′) − f(z)| = |∫_{γ} \frac{df}{dz_{1}} dz_{1}| ≤ K · (length γ).

Since this holds in particular when γ is the shortest path, with length |z′−z|, the claimed inequality holds in this case.

The n > 1 case follows from the n = 1 case:

∑_{i=1}^{n} |f_{i}(z ′) − f_{i}(z⃗)| = ∑_{i=1}^{n} |f_{i}(z′_{1}, …, z′_{n}) − f_{i}(z′_{1}, …, z′_{n−1}, z_{n})

+f_{i}(z′_{1}, …, z′_{n−1}, z_{n}) − f_{i}(z′_{1}, …, z′_{n−2}, z_{n−1}, z_{n})

+ … + f_{i}(z′_{1}, z_{2}, …, z_{n}) − f_{i}(z_{1}, …, z_{n})|
≤ ∑_{i=1}^{n} ∑_{j=1}^{n} ( sup_{z⃗∈𝔻_{r}} |\frac{df_{i}}{dz_{j}} (z⃗)|) |z′_{j} − z_{j}|

= ∑_{j=1}^{n} (∑_{i=1}^{n} sup_{z⃗∈𝔻_{r}} |\frac{df_{i}}{dz_{j}} (z⃗)|) |z′_{j} − z_{j}|

≤ ∑_{j=1}^{n} ( max_{l=1,...,n} \{∑_{i=1}^{n} sup_{z⃗∈𝔻_{r}} |\frac{df_{i}}{dz_{l}} (z⃗)|\}) |z′_{j} − z_{j}|

≤ K ∑_{j=1}^{n} |z′_{j} − z_{j}|.

Theorem 3.4 showed that an approximate solution of (8), (9) exists; the estimates from Corollary 3.5 will measure how close the approximation is to an actual solution. Substituting the normalized solution p⃗ from the Theorem into e_{2}, e_{3} in the RHS of (8), (9) gives two quantities depending on z, z̄:

q_{2}(z, z̄) = e_{2}(z, z̄) + p_{2}(z, z̄^{2} + e_{2}, zz̄ + e_{3})
−(2z̄\overline{p_{1}(z, z̄^{2} + e_{2})} + \overline{p_{1}(z, z̄^{2} + e_{2})}^{2})
= p_{2}^{E}(z, z̄^{2} + e_{2}) − p_{2}^{E}(z, z̄^{2})  (12)
+(zz̄ + e_{3})p_{2}^{O}(z, z̄^{2} + e_{2}) − zz̄p_{2}^{O}(z, z̄^{2})
−2z̄\overline{(p_{1}(z, z̄^{2} + e_{2}) − p_{1}(z, z̄^{2}))} − \overline{p_{1}(z, z̄^{2} + e_{2})}^{2},
q_{3}(z, z̄) = e_{3}(z, z̄) + p_{3}(z, z̄^{2} + e_{2}, zz̄ + e_{3})
−z\overline{p_{1}(z, z̄^{2} + e_{2})} − z̄p_{1}(z, z̄^{2} + e_{2}) − |p_{1}(z, z̄^{2} + e_{2})|^{2}
= p_{3}^{E}(z, z̄^{2} + e_{2}) − p_{3}^{E}(z, z̄^{2})  (13)
+(zz̄ + e_{3})p_{3}^{O}(z, z̄^{2} + e_{2}) − zz̄p_{3}^{O}(z, z̄^{2})

−z\overline{(p_{1}(z, z̄^{2} + e_{2}) − p_{1}(z, z̄^{2}))} − z̄(p_{1}(z, z̄^{2} + e_{2}) − p_{1}(z, z̄^{2}))

−|p_{1}(z, z̄^{2} + e_{2})|^{2}.

If p⃗(z, z_{2}, z_{3}) is complex analytic on Δ_{r}, and σ^{2} + |e⃗|_{σ} < r^{2}, then q⃗ = (q_{2}(z, z̄), q_{3}(z, z̄)) is a real analytic function for |z| < σ. If q⃗(z, z̄) happens to be identically zero, the manifold M has been brought to normal form by the functions p⃗; if not, the degree of q⃗ is at least 2d − 2 as mentioned previously, and its values can be bounded in terms of e⃗. Define q⃗(z, ζ) = (q_{2}(z, ζ), q_{3}(z, ζ)) by (12), (13), with ζ formally substituted for z̄.

Theorem 4.2. There are some constants c_{2} > 0 and δ_{1} > 0 such that if \frac{1}{2} < σ < r ≤ 1, and e⃗ is as in Theorem 3.4, with |e⃗|_{r} ≤ δ_{1}(r − σ), then

|q⃗|_{σ} ≤ \frac{c_{2}|e⃗|_{r}^{2}}{r − σ} .

Proof. Note that if δ_{1} ≤ 1, the formal series for q⃗ is in fact convergent on D_{σ}, since then |ζ^{2} + e_{2}(z, ζ)| < σ^{2} + δ_{1}(r − σ) < σ^{2} + (r − σ)(r + σ) = r^{2}, and p⃗ is convergent on Δ_{r} by Theorem 3.4. To find a bound on the norm of q⃗, the following corollary of the n = 1 case of Lemma 4.1 will apply:

If 0 < R_{2} < R_{1} and f(z) is holomorphic for |z| < R_{1}, and g : X → ℂ is any function bounded by M ≤ R_{1} − R_{2}, then for (z, x) ∈ {|z| < R_{2}} × X,

|f(z + g(x)) − f(z)| ≤ M sup_{|z|<R_{1}} |\frac{df}{dz} | .

Using this estimate and the bounds for derivatives from Corollary 3.5, with R_{1} = ρ = \frac{1}{2}(σ + r), R_{2} = σ, g = e_{2}, and M = |e_{2}|_{σ} ≤ δ_{1}(r − σ) ≤ ρ − σ for δ_{1} ≤ \frac{1}{2} gives the following inequalities:

|q_{2}|_{σ} ≤ |e_{2}|_{σ} ∥\frac{dp_{2}^{E}}{dz_{2}} ∥_{ρ} + σ^{2}|e_{2}|_{σ} ∥\frac{dp_{2}^{O}}{dz_{2}} ∥_{ρ}

+|e_{3}|_{σ}∥p_{2}^{O}∥_{ρ} + 2σ|e_{2}|_{σ} ∥\frac{dp_{1}}{dz_{2}} ∥_{ρ} + ∥p_{1}∥_{ρ}^{2}

≤ |e_{2}|_{σ} \frac{|e_{2}|_{r}}{r^{2} − ρ^{2}} + \frac{σ^{2}}{r^{2}} |e_{2}|_{σ} \frac{2(|e_{2}|_{r} + |e_{3}|_{r})}{r^{2} − ρ^{2}}

+\frac{2}{r^{2}} |e_{3}|_{σ}(|e_{2}|_{r} + |e_{3}|_{r}) + 2\frac{σ}{r} |e_{2}|_{σ} \frac{|e_{3}|_{r}}{r^{2} − ρ^{2}} + (\frac{1}{2r} |e_{2}|_{r} + \frac{1}{r}|e_{3}|_{r})^{2} ,

|q_{3}|_{σ} ≤ |e_{2}|_{σ} ∥\frac{dp_{3}^{E}}{dz_{2}} ∥_{ρ} + σ^{2}|e_{2}|_{σ} ∥\frac{dp_{3}^{O}}{dz_{2}} ∥_{ρ}

+|e_{3}|_{σ}∥p_{3}^{O}∥_{ρ} + 2σ|e_{2}|_{σ} ∥\frac{dp_{1}}{dz_{2}} ∥_{ρ} + ∥p_{1}∥_{ρ}^{2}

≤ |e_{2}|_{σ} \frac{2|e_{3}|_{r} + \frac{1}{2}|e_{2}|_{r}}{r^{2} − ρ^{2}} + \frac{σ^{2}}{r^{2}} |e_{2}|_{σ} \frac{2|e_{3}|_{r} + \frac{1}{2}|e_{2}|_{r}}{r^{2} − ρ^{2}}

+|e_{3}|_{σ} \frac{4|e_{3}|_{r} + |e_{2}|_{r}}{2r^{2}} + 2\frac{σ}{r} |e_{2}|_{σ} \frac{|e_{3}|_{r}}{r^{2} − ρ^{2}} + (\frac{1}{2r} |e_{2}|_{r} + \frac{1}{r}|e_{3}|_{r})^{2} .

The purpose of constructing the functions p⃗ was to define a new coordinate system. To see how close the defining equations in the new coordinates are to the normal form equations, some more estimates will be needed. In particular, the new coordinate system will only be defined on some polydisc Δ, and the new equations z̃_{2} = ¯ z̃^{2} + ẽ_{2}(z̃, ¯ z̃), z̃_{3} = z̃¯z̃ + ẽ_{3}(z̃, ¯ z̃) will only be defined for z̃ in some disc of radius r̃.

The following Lemma, the “standard iteration procedure” for inverse functions, ([M], [SM] §§26, 33), will be used twice, in the construction of both the new coordinate system and the new defining equations. It is stated in general, with a sketch of a proof.

Lemma 4.3. Suppose 0 < R_{2}^{i} < R_{1}^{i} for i = 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} \{∑_{i=1}^{n} sup_{z⃗∈𝔻^{1}} |\frac{df_{i}}{dz_{j}} (z⃗)|\} ≤ K < 1,

and

∑_{i=1}^{n} sup_{z⃗∈𝔻^{2}} |f_{i}(z⃗)| ≤ (1 − K) min_{i=1,...,n}{R_{1}^{i} − R_{2}^{i}}.

Then, given w⃗ ∈ 𝔻^{2}, there exists a unique solution z⃗ ∈ 𝔻^{1} of the equation

w⃗ = z⃗ + f(z⃗),

and this solution satisfies

∑_{i=1}^{n} |z_{i} − w_{i}| ≤ \frac{1}{1 − K} ∑_{i=1}^{n} |f_{i}(w⃗)|.

Proof. The idea is to construct a sequence z ^{(k)} = (z_{1}^{(k)}, …, z_{n}^{(k)}), k = 0, 1, 2, …, converging to z⃗ so that each z ^{(k)} ∈ 𝔻^{1}. Start with an initial approximate solution z ^{(0)} = w⃗. Inductively, assume z ^{(m)} ∈ 𝔻^{1} for 0 ≤ m ≤ k, and define

z ^{(k+1)} = w⃗ − f(z ^{(k)})  (14)

Then, z ^{(k+1)} ∈ 𝔻^{1}, and for k ≥ 1, Lemma 4.1 applies:

∑_{i=1}^{n} |z_{i}^{(k+1)} − z_{i}^{(k)}| = ∑_{i=1}^{n} |f_{i}(z ^{(k)}) − f_{i}(z ^{(k−1)})|

≤ K ∑_{j=1}^{n} |z_{j}^{(k)} − z_{j}^{(k−1)}|

≤ K^{k} ∑_{j=1}^{n} |z_{j}^{(1)} − z_{j}^{(0)}| = K^{k} ∑_{j=1}^{n} |f_{j}(w⃗)|.

Since K < 1, it follows that z ^{(k)} forms a Cauchy sequence, and that

∑_{i=1}^{n} |z_{i}^{(k)} − w_{i}| ≤ ∑_{i=1}^{n}(|z_{i}^{(k)} − z_{i}^{(k−1)}| + |z_{i}^{(k−1)} − z_{i}^{(k−2)}| + ⋯ + |z_{i}^{(1)} − w_{i}|)

≤ (K^{k−1} + ⋯ + K + 1) ∑_{i=1}^{n} |f_{i}(w⃗)|

= \frac{1 − K^{k}}{1 − K} ∑_{i=1}^{n} |f_{i}(w⃗)| ≤ min_{i=1,...,n}\{R_{1}^{i} − R_{2}^{i}\},

so z⃗ = lim_{k→∞} z ^{(k)} exists, with |z_{i} − w_{i}| ≤ R_{1}^{i} − R_{2}^{i}, and so z⃗ ∈ 𝔻^{1}, and z⃗

satisfies the claimed estimate. To see that z⃗ is a solution of the original equation, take the k → ∞ limit of equation (14).

Finally, to show that if z⃗ is unique, suppose z ′ is another solution; then by Lemma 4.1 again,

∑_{i=1}^{n} |z_{i} − z′_{i}| = ∑_{i=1}^{n} |f_{i}(z⃗) − f_{i}(z ′)| ≤ K ∑_{j=1}^{n} |z_{j} − z′_{j}|,

contradicting K < 1. In fact, z⃗ is a holomorphic function of w⃗ on the polydisc 𝔻^{2}.

Theorem 4.4. There is some constant δ_{2} > 0 so that for any radii \frac{1}{2} < σ < r ≤ 1, and e⃗, p⃗ as in Theorem 3.4, with |e⃗|_{r} ≤ δ_{2}(r − σ) and ρ = \frac{1}{2}(r + σ), the transformation

Ψ : (z, z_{2}, z_{3}) ↦ (z̃, z̃_{2}, z̃_{3}) = (z + p_{1}, z_{2} + p_{2}, z_{3} + p_{3})

has a holomorphic inverse ψ(z̃, z̃_{2}, z̃_{3}) = (z, z_{2}, z_{3}) such that if (z̃, z̃_{2}, z̃_{3}) ∈ Δ_{σ}, then ψ(z̃, z̃_{2}, z̃_{3}) ∈ Δ_{ρ}.

Proof. By Corollary 3.5,

max_{j=1,2,3} \{∑_{i=1}^{3} ∥\frac{dp_{i}}{dz_{j}} ∥_{ρ}\} ≤ \frac{c_{1}|e⃗|_{r}}{r − ρ} ≤ \frac{c_{1}δ_{2}(r − σ)}{r − ρ} = 2δ_{2}c_{1} ≤ \frac{1}{2} = K,

if δ_{2} ≤ \frac{1}{4c_{1}} . Also by Corollary 3.5,

∥p_{1}∥_{σ} + ∥p_{2}∥_{σ} + ∥p_{3}∥_{σ} ≤ 3c_{1}|e⃗|_{r} ≤ 3c_{1}δ_{2}(r − σ) ≤ (1 − K)(ρ − σ),

if δ_{2} ≤ \frac{ρ−σ}{6c_{1}(r−σ)} = \frac{1}{12c_{1}} . The hypotheses of Lemma 4.3 are satisfied, so given (z̃, z̃_{2}, z̃_{3}) ∈ Δ_{σ}, there exists a unique (z, z_{2}, z_{3}) ∈ Δ_{ρ} such that (z̃, z̃_{2}, z̃_{3}) = (z + p_{1}, z_{2} + p_{2}, z_{3} + p_{3}). This defines ψ so that Ψ ∘ ψ is the identity map on Δ_{σ}.

Theorem 4.5. There is some constant δ_{3} > 0 so that for any radii \frac{1}{2} < r′ < r ≤ 1, and e⃗, p⃗ as in Theorem 3.4, with |e⃗|_{r} ≤ δ_{3}(r−r′), and σ = r′+\frac{1}{3}(r−r′), the transformation

(z, ζ) ↦ (z̃, ζ̃) = (z + p_{1}(z, ζ^{2} + e_{2}(z, ζ)), ζ + \overline{p_{1}(ζ̄, z̄^{2} + e_{2}(ζ̄, z̄))}) (15)

has a holomorphic inverse ϕ(z̃, ζ̃) = (z, ζ) such that if (z̃, ζ̃) ∈ D_{r′}, then ϕ(z̃, ζ̃) ∈ D_{σ}.

Proof. Let ρ = r′ + \frac{2}{3}(r − r′), so σ − r′ = ρ − σ = r − ρ = \frac{1}{3}(r − r′). If (z, ζ) ∈ D_{σ}, and δ_{3} ≤ \frac{2}{3}, then |e⃗|_{σ} ≤ r^{2} − σ^{2}, and the map (15) is welldefined and holomorphic on D_{σ}. Omitting the details, Cauchy’s estimate can be used in calculations similar to those in Corollary 3.5, so that there is some constant c_{3} > 0 (not depending on e⃗), with

max \{|\frac{d}{dz} p_{1}(z, ζ^{2} + e_{2}(z, ζ))|_{σ} + |\frac{d}{dz} \overline{p_{1}(ζ̄, z̄^{2} + e_{2}(ζ̄, z̄))}|_{σ} ,

|\frac{d}{dζ} p_{1}(z, ζ^{2} + e_{2}(z, ζ))|_{σ} + |\frac{d}{dζ} \overline{p_{1}(ζ̄, z̄^{2} + e_{2}(ζ̄, z̄))}|_{σ}\}

≤ \frac{c_{3}|e⃗|_{r}}{r − ρ} ≤ \frac{c_{3}δ_{3}(r − r′)}{r − ρ} ≤ \frac{1}{2},

if δ_{3} ≤ \frac{r−ρ}{2c_{3}(r−r′)} = \frac{1}{6c_{3}} . It follows from Corollary 3.5 that

|p_{1}(z, ζ^{2} + e_{2}(z, ζ))|_{r′} + |\overline{p_{1}(ζ̄, z̄^{2} + e_{2}(ζ̄, z̄))}|_{r′}
≤ 2∥p_{1}(z, z_{2})∥_{r} ≤ 2c_{1}|e⃗|_{r} ≤ 2c_{1}δ_{3}(r − r′) ≤ \frac{1}{2}(σ − r′),

if δ_{3} ≤ \frac{σ−r′}{4c_{1}(r−r′)} = \frac{1}{12c_{1}} . So, by Lemma 4.3, given (z̃, ζ̃) ∈ D_{r′}, there exists a unique (z, ζ) ∈ D_{σ} such that

(z̃, ζ̃) = (z + p_{1}(z, ζ^{2} + e_{2}(z, ζ)), ζ + \overline{p_{1}(ζ̄, z̄^{2} + e_{2}(ζ̄, z̄))}).

By inspection of the form of (15), if (z, ζ) ∈ D_{σ}, and (z, ζ) ↦ (z̃, ζ̃), then (ζ̄, z̄) ↦ (\overline{ζ̃}, ¯ z̃). If, further, (z̃, ζ̃) = (\overline{ζ̃}, ¯ z̃) ∈ D_{r′}, then (z, ζ) = (ζ̄, z̄) by uniqueness of the inverse. In particular, if |z̃| < r′, then ϕ(z̃, ¯ z̃) is of the form (z, z̄) for some z with |z| < σ. Also, if there exists some other z′ such that |z′| < σ and z̃ = z′ + p_{1}(z′, (z̄′)^{2} + e_{2}(z′, z̄′)), then

¯z̃ = \overline{z′ + p_{1}(z′, (z̄′)^{2} + e_{2}(z′, z̄′))}
= z̄′ + \overline{p_{1}(\overline{z̄′}, (z̄′)^{2} + e_{2}(\overline{z̄′}, z̄′))},

so ϕ(z̃, ¯ z̃) = (z′, z̄′) = (z, z̄) by uniqueness.

Theorem 4.6. There exist constants c_{4} > 0 and δ_{4} > 0 such that for any \frac{1}{2} < r′ < r ≤ 1 (with σ, ρ as in the previous Theorem), and any e⃗ as in Theorem 3.4 with |e⃗|_{r} ≤ δ_{4}(r − r′), there exist a biholomorphic map Ψ : (z, z_{2}, z_{3}) ↦ (z̃, z̃_{2}, z̃_{3}), with the domain of ψ = Ψ^{−1} containing Δ_{σ}, and holomorphic functions ẽ_{2}, ẽ_{3}, with domain containing D_{r′}, such that the defining equations for M are

z̃_{2} = ¯ z̃^{2} + ẽ_{2}(z̃, ¯ z̃),
z̃_{3} = z̃¯z̃ + ẽ_{3}(z̃, ¯ z̃),

for |z̃| < r′. Further, the degree of ẽ = (ẽ_{2}, ẽ_{3}) is at least 2d − 2, and

|ẽ|_{r′} ≤ \frac{c_{4}|e⃗|_{r}^{2}}{r − r′} .

Proof. Initially, choose δ_{4} ≤ min\{\frac{2}{3}δ_{1}, \frac{2}{3}δ_{2}, δ_{3}\}, so that Theorems 4.2, 4.4, 4.5 apply, and define Ψ, ψ, q⃗, and ϕ in terms of the given e⃗ and the functions p⃗ constructed in Theorem 3.4. Define ẽ(z̃, ζ̃) = (q⃗ ∘ ϕ)(z̃, ζ̃), so that ẽ is a pair of compositions of holomorphic functions, with domain containing D_{r′}, and

|ẽ|_{r′} ≤ |q⃗|_{σ} ≤ \frac{c_{2}|e⃗|_{r}^{2}}{r − σ} = \frac{c_{2}|e⃗|_{r}^{2}}{\frac{2}{3}(r − r′)}.

Since ϕ(z̃, ¯ z̃) has no constant terms, and q⃗ has degree ≥ 2d − 2 by construction, ẽ also has degree at least 2d − 2.

Suppose |z̃| < r′, and define z̃_{2} = ¯ z̃^{2} + ẽ_{2}(z̃, ¯ z̃), and z̃_{3} = z̃¯z̃ + ẽ_{3}(z̃, ¯ z̃). The claim of the Theorem is that ψ(z̃, z̃_{2}, z̃_{3}) ∈ M.

If δ_{4}^{2} ≤ \frac{2}{9c_{2}} , then |ẽ|_{r′} ≤ \frac{c_{2}(δ_{4}(r−r′))^{2}}{r−σ} ≤ \frac{1}{3}(r − r′) ≤ σ^{2} − (r′)^{2}, so (z̃, z̃_{2}, z̃_{3}) ∈ Δ_{σ}.

By Theorem 4.5, there exists a unique z such that |z| < σ and z̃ = z + p_{1}(z, z̄^{2} + e_{2}(z, z̄)). (This z is the first component of (z, z̄) = ϕ(z̃, ¯ z̃).) Then, define z_{2} = z̄^{2} +e_{2}(z, z̄), z_{3} = zz̄+e_{3}(z, z̄); since |z| < σ < r, (z, z_{2}, z_{3}) ∈ M, and if δ_{4} ≤ \frac{1}{3} < \frac{ρ^{2}−σ^{2}}{3(ρ−σ)} = \frac{ρ^{2}−σ^{2}}{r−r′}, then |z_{2}| < σ^{2}+δ_{4}(r−r′) < ρ^{2}, and similarly |z_{3}| < ρ^{2}, so (z, z_{2}, z_{3}) ∈ Δ_{ρ}.

Ψ(z, z_{2}, z_{3}) = (z + p_{1}(z, z̄^{2} + e_{2}(z, z̄)),
z̄^{2} + e_{2}(z, z̄) + p_{2}(z, z̄^{2} + e_{2}(z, z̄), zz̄ + e_{3}(z, z̄)),
zz̄ + e_{3}(z, z̄) + p_{3}(z, z̄^{2} + e_{2}(z, z̄), zz̄ + e_{3}(z, z̄)))
= (z̃, \overline{(z̃ − p_{1})}^{2} + e_{2} + p_{2}, (z̃ − p_{1})\overline{(z̃ − p_{1})} + e_{3} + p_{3})
= (z̃, ¯ z̃^{2} + q_{2}(z, z̄), z̃¯z̃ + q_{3}(z, z̄))
= (z̃, ¯ z̃^{2} + q_{2}(ϕ(z̃, ¯ z̃)), z̃¯z̃ + q_{3}(ϕ(z̃, ¯ z̃)))
= (z̃, z̃_{2}, z̃_{3}),

by construction of p⃗, q⃗, and z̃_{2}, z̃_{3}. By the uniqueness of Theorem 4.4, ψ(z̃, z̃_{2}, z̃_{3}) = (z, z_{2}, z_{3}) ∈ M.

5 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 few technical lemmas will be needed to control the behavior of composite mappings. The main result, Theorem 5.7, uses these lemmas and the estimates of the previous Section to prove convergence of a sequence of transformations, following the ideas of [M] and [SM] §§26, 32.

Notation 5.1. For a 3×3 matrix of complex-valued functions F = (F_{ij}(z⃗)) on Δ_{r}, define

|||F |||_{r} = max_{j=1,2,3} {∑_{i=1}^{3} sup_{z⃗∈Δ_{r}} |F_{ij}(z⃗)|} .

This “maximum column sum” norm appeared already, in Corollary 3.5 and Lemmas 4.1, 4.3, in the case where F = Df = Dz⃗f, the Jacobian matrix of some map f : Δ_{r} → ℂ^{3} at z⃗ ∈ Δ_{r}.

Lemma 5.2. If |||A|||_{r} < 1, then I + A is invertible (where I is the 3 × 3 identity matrix), and

|||(I + A)^{−1}|||_{r} ≤ \frac{1}{1 − |||A|||_{r}} .

Proof. It is enough (cf [HJ]) to check that |||I|||_{r} = 1 and that this norm is “submultiplicative,” |||F · G|||_{r} ≤ |||F |||_{r} · |||G|||_{r}. The following calculation is similar to the steps of Lemma 4.1, and generalizes to 𝔻_{r} ⊆ ℂ^{n}.

|||F · G|||_{r} = max_{j=1,2,3} {∑_{i=1}^{3} sup_{z⃗∈Δ_{r}} |∑_{k=1}^{3} F_{ik}(z⃗) · G_{kj}(z⃗)|}

≤ max_{j=1,2,3} {∑_{i=1}^{3} sup_{z⃗∈Δ_{r}} ∑_{k=1}^{3} |F_{ik}(z⃗)| · |G_{kj}(z⃗)|}

≤ max_{j=1,2,3} {∑_{i=1}^{3} ∑_{k=1}^{3} ( sup_{z⃗∈Δ_{r}} |F_{ik}(z⃗)|) · ( sup_{z⃗∈Δ_{r}} |G_{kj}(z⃗)|)}

= max_{j=1,2,3} {∑_{k=1}^{3} (∑_{i=1}^{3} sup_{z⃗∈Δ_{r}} |F_{ik}(z⃗)|) · ( sup_{z⃗∈Δ_{r}} |G_{kj}(z⃗)|)}

≤ max_{j=1,2,3} {∑_{k=1}^{3} ( max_{l=1,2,3} {∑_{i=1}^{3} sup_{z⃗∈Δ_{r}} |F_{il}(z⃗)|})·( sup_{z⃗∈Δ_{r}} |G_{kj}(z⃗)|)}
= |||F |||_{r} · |||G|||_{r}

Also, the following elementary fact from the calculus of one real variable will be used.

Lemma 5.3. If a_{i} is a sequence such that 0 ≤ a_{i} < 1 and ∑_{i=0}^{∞} a_{i} is a

convergent series, then the sequence of partial products

∏_{i=0}^{N} \frac{1}{1 − a_{i}}

is bounded above by some positive limit.

The following notation will be convenient.

Notation 5.4. For ν = 0, 1, 2, …, define a sequence \{1, \frac{3}{4}, \frac{4}{6}, \frac{5}{8}, …\} 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{r_{ν+1} − r_{ν+2}}{r_{ν} − r_{ν+1}} = \frac{ν + 1}{ν + 3} ≥ \frac{1}{3}.

Notation 5.5. Define σ_{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 4.5.

Recall that given η > 0, there is some scaling transformation so that M ∩ Δ_{1} is defined by equations (2), (3), with e⃗ holomorphic on D_{1}, degree d ≥ 3, and |e⃗|_{1} ≤ η.

Notation 5.6. Denote e⃗_{0} = e⃗ (so |e⃗_{0}|_{r_{0}} = |e⃗|_{1} ≤ η), and inductively define the formal series e⃗_{ν+1}(z, ζ) in terms of e⃗_{ν}(z, ζ), by the e⃗ ↦ ẽ procedure of Theorem 4.6, 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} − 2; it can be checked that this, together with d_{0} = d ≥ 3, implies d_{ν} ≥ 2^{ν} + 2.

The plan is to show that the estimate in the hypothesis of Theorem 4.6 holds for all ν, to get a sequence of transformations ψ_{ν} : Δ_{σ_{ν}} → Δ_{ρ_{ν}}, so that the composition ψ_{0} ∘ … ∘ ψ_{ν−1} ∘ ψ_{ν} : Δ_{σ_{ν}} → Δ_{ρ_{0}} is well-defined, e⃗_{ν} is holomorphic on D_{r_{ν}}, and

lim_{ν→∞} |e⃗_{ν}|_{r_{ν}} = 0.

Theorem 5.7. There exists η > 0 so that if e⃗_{0} and M are as described above, then there exists a transformation Ψ : Δ_{1} → ℂ^{3}, which has a holomorphic inverse ψ : Δ_{\frac{1}{2}} → Δ_{1}, and such that if (z̃, z̃_{2}, z̃_{3}) ∈ M̃ ∩ Δ_{\frac{1}{2}} , then ψ(z̃, z̃_{2}, z̃_{3}) ∈ M.

Proof. Let δ_{5} = min\{δ_{4}, \frac{1}{3c_{4}} \}. It will be shown that

|e⃗_{ν}|_{r_{ν}} ≤ δ_{5}(r_{ν} − r_{ν+1}) ⟹ |e⃗_{ν+1}|_{r_{ν+1}} ≤ δ_{5}(r_{ν+1} − r_{ν+2}).

By Theorem 4.6, |e⃗_{ν}|_{r_{ν}} ≤ δ_{4}(r_{ν} − r_{ν+1}) and |e⃗_{ν}|_{r_{ν}} ≤ \frac{1}{3c_{4}} (r_{ν} − r_{ν+1}) imply

|e⃗_{ν+1}|_{r_{ν+1}} ≤ \frac{c_{4}|e⃗_{ν}|_{r_{ν}}^{2}}{r_{ν} − r_{ν+1}} ≤ \frac{1}{3}|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}{3}|e⃗_{ν}|_{r_{ν}} ≤ \frac{1}{3}δ_{5}(r_{ν} − r_{ν+1}) ≤ δ_{5}(r_{ν+1} − r_{ν+2}),

which proves the claimed implication. Using this as an inductive step, and starting the induction with |e⃗_{0}|_{r_{0}} ≤ δ_{5}(r_{0} − r_{1}) = \frac{1}{4}δ_{5} = η, the hypothesis of Theorem 4.6 is satisfied for all ν. One conclusion from Theorem 4.6 is that e⃗_{ν} is holomorphic on D_{ν}, with degree d_{ν} ≥ 2^{ν} + 2, and |e⃗_{ν}|_{r_{ν}} ≤ 3^{−ν}η. The other conclusion is that Ψ_{ν} ∘ … ∘ Ψ_{0} is holomorphic, with inverse ψ_{0} ∘ … ∘ ψ_{ν} : Δ_{σ_{ν}} → Δ_{ρ_{0}} such that if |z̃| < r_{ν+1}, and z̃_{2} = ¯ z̃^{2} + (e⃗_{ν+1})_{2}(z̃, ¯ z̃), and z̃_{3} = z̃¯z̃+(e⃗_{ν+1})_{3}(z̃, ¯ z̃), then (ψ_{0}∘…∘ψ_{ν})(z̃, z̃_{2}, z̃_{3}) ∈ M. For (z, z_{2}, z_{3}) ∈ Δ_{\frac{1}{2}} , the sequence (depending on ν) (ψ_{0} ∘ … ∘ ψ_{ν−1} ∘ ψ_{ν})(z, z_{2}, z_{3}) is contained in Δ_{1}. The following argument, beginning with several applications of Lemma 4.1, shows this sequence is a Cauchy sequence, and converges to some value ψ(z, z_{2}, z_{3}).

∑_{i=1}^{3} |(ψ_{0} ∘ … ∘ ψ_{ν+1})_{i}(z, z_{2}, z_{3}) − (ψ_{0} ∘ … ∘ ψ_{ν})_{i}(z, z_{2}, z_{3})|

= ∑_{i=1}^{3} |(ψ_{0})_{i}((ψ_{1}∘ … ∘ ψ_{ν+1})(z, z_{2}, z_{3})) − (ψ_{0})_{i}((ψ_{1}∘ … ∘ ψ_{ν})(z, z_{2}, z_{3}))|

≤ |||Dψ_{0}|||_{ρ_{1}} ·∑_{j=1}^{3} |(ψ_{1} ∘ … ∘ ψ_{ν+1})_{j}(z, z_{2}, z_{3}) − (ψ_{1} ∘ … ∘ ψ_{ν})_{j}(z, z_{2}, z_{3})|

≤ ( ∏_{m=0}^{ν} |||Dψ_{m}|||_{ρ_{m+1}}) · ∑_{j=1}^{3} |(ψ_{ν+1})_{j}(z, z_{2}, z_{3}) − z_{j}|.  (16)

By the estimate from Lemma 4.3, with f = p⃗_{ν+1} and K = \frac{1}{2} from the proof of Theorem 4.4, and then using the bound for p⃗ from Corollary 3.5,

∑_{j=1}^{3} |(ψ_{ν+1})_{j}(z, z_{2}, z_{3}) − z_{j}| ≤ \frac{1}{1 − \frac{1}{2}} ∑_{j=1}^{3} |(p⃗_{ν+1})_{j}(z, z_{2}, z_{3})|

≤ 2 ∑_{j=1}^{3} ∥(p⃗_{ν+1})_{j}∥_{\frac{1}{2}}

≤ 6c_{1}|e⃗_{ν+1}|_{r_{ν+1}} ≤ \frac{6c_{1}η}{3^{ν+1}} .

It follows from Dz⃗ψ_{m} = (I + D_{ψ_{m}(z⃗)}p⃗_{m})^{−1} and Lemma 5.2 that:

|||Dψ_{m}|||_{ρ_{m+1}} = |||(I + D_{ψ_{m}(z⃗)}p⃗_{m})^{−1}|||_{ρ_{m+1}}
≤ |||(I + Dp⃗_{m})^{−1}|||_{ρ_{m}}
≤ \frac{1}{1 − |||Dp⃗_{m}|||_{ρ_{m}}} .

Then, by Lemma 5.3, the product from (16) is bounded above by some constant c_{5}, since by Corollary 3.5,

∑_{m=0}^{∞} |||Dp⃗_{m}|||_{ρ_{m}} ≤ ∑_{m=0}^{∞} \frac{c_{1}|e⃗_{m}|_{r_{m}}}{r_{m} − ρ_{m}} ≤ ∑_{m=0}^{∞} \frac{2(m + 1)(m + 2)c_{1}η}{3^{m−1}} ,

a convergent infinite series.

The inequality

∑_{i=1}^{3} |(ψ_{0} ∘ … ∘ ψ_{ν+1})_{i}(z, z_{2}, z_{3}) − (ψ_{0} ∘ … ∘ ψ_{ν})_{i}(z, z_{2}, z_{3})| ≤ \frac{2c_{1}c_{5}η}{3^{ν}}

is enough to show that the sequence of composite functions converges pointwise and uniformly to a function ψ on Δ_{\frac{1}{2}} .

6 Stabilizer of the normal form

As an appendix to the main result, the set of formal transformations of ℂ^{3} which fix the origin and preserve the defining equations of M̃ is constructed. This is intended to motivate the choice of normalization in Section 3.

For z⃗ = (z, z_{2}, z_{3}), define (z̃, z̃_{2}, z̃_{3}) = Ψ(z⃗) = Az⃗ + (p_{1}, p_{2}, p_{3}), as in equations (5), (6), (7), but with a 3 × 3 constant matrix A of complex coefficients on the linear terms. Since the linear part of Ψ should fix the z-axis, the tangent plane of M̃, the matrix A is of the form

A = ( \begin{matrix}a_{11} & a_{12} & a_{13} \\ 0 & a_{22} & a_{23} \\ 0 & a_{32} & a_{33} \end{matrix} ) .

The a_{12}, a_{13} entries are coefficients on terms of the form z̃ = z+a_{12}z_{2}+a_{13}z_{3}, and such a “weight 2” transformation does not contribute quadratic terms to the new defining equations, and might be more conveniently considered as part of the p_{1}(z, z_{2}, z_{3}) function. The only matrix entries which affect the quadratic terms of the defining equations are a_{11} and the a_{22}, a_{23}, a_{32}, a_{33} block, and since Ψ should be (formally) invertible, det A ≠ 0, so a_{11} can be any nonzero s ∈ ℂ, and a_{22}a_{33} − a_{23}a_{32} ≠ 0. If Ψ stabilizes the defining equations, z_{2} = z̄^{2}, z_{3} = zz̄, then z̃_{2} = ¯ z̃^{2} and z̃_{3} = z̃¯z̃, so

a_{22}z_{2} + a_{23}z_{3} + p_{2} = \overline{(sz + p_{1})}^{2}
= a_{22}z̄^{2} + a_{23}zz̄ + p_{2},
a_{32}z_{2} + a_{33}z_{3} + p_{3} = (sz + p_{1})\overline{(sz + p_{1})}
= a_{32}z̄^{2} + a_{33}zz̄ + p_{3}.

Comparing the quadratic coefficients gives a_{22} = s̄^{2}, a_{23} = a_{32} = 0, and a_{33} = ss̄, so Ψ is of the form

(z, z_{2}, z_{3}) ↦ (sz, s̄^{2}z_{2}, |s|^{2}z_{3}) + (p_{1}, p_{2}, p_{3}),

with p_{1} weight ≥ 2, and p_{2}, p_{3} weight ≥ 3. Constructing q⃗ as in Section 4 gives:

z̃_{2} − ¯ z̃^{2} = p_{2}(z, z̄^{2}, zz̄) − 2s̄z̄\overline{p_{1}(z, z̄^{2}, zz̄)} − (\overline{p_{1}(z, z̄^{2}, zz̄)})^{2},
z̃_{3} − z̃¯z̃ = p_{3}(z, z̄^{2}, zz̄) − sz\overline{p_{1}(z, z̄^{2}, zz̄)} − s̄z̄p_{1}(z, z̄^{2}, zz̄) − |p_{1}(z, z̄^{2}, zz̄)|^{2}.

Setting RHS equal to 0 determines p_{2}(z, z̄^{2}, zz̄) and p_{3}(z, z̄^{2}, zz̄) in terms of p_{1}, however, p_{1} cannot be arbitrary, and p_{2}(z, z_{2}, z_{3}), p_{3}(z, z_{2}, z_{3}) are not uniquely defined by their restriction to M̃.

Let Q be the subalgebra of ℂ[[z, z̄]] of formal series of monomials of the form z^{a}(z̄^{2})^{k}(zz̄)^{c}; any monomial can be written in this form except z̄^{b} with b odd.

Let J denote the ideal generated by (z^{2}z_{2} − z_{3}^{2}) in ℂ[[z, z_{2}, z_{3}]]. As mentioned in the proof of Theorem 3.4, any formal series p can be written as (z^{2}z_{2} − z_{3}^{2}) · Q + z_{3}p^{O}(z, z_{2}) + p^{E}(z, z_{2}). A series in ℂ[[z, z̄]] is in Q if and only if it is the restriction of some z_{3}p^{O}(z, z_{2}) + p^{E}(z, z_{2}) to z_{2} = z̄^{2}, z̄_{3} = zz̄.

In order for p_{1}, p_{2}, p_{3} to satisfy:

p_{2}(z, z̄^{2}, zz̄) = 2s̄z̄\overline{p_{1}(z, z̄^{2}, zz̄)} + (\overline{p_{1}(z, z̄^{2}, zz̄)})^{2},
p_{3}(z, z̄^{2}, zz̄) = sz\overline{p_{1}(z, z̄^{2}, zz̄)} + s̄z̄p_{1}(z, z̄^{2}, zz̄) + |p_{1}(z, z̄^{2}, zz̄)|^{2},

the RHS must be in Q (since the LHS is in Q). In particular, p̄_{1} · (2s̄z̄+ p̄_{1}) and p_{1} · (s̄z̄ + p̄_{1}) must both be in Q when p_{1} is restricted to M̃. Any p_{1} satisfying this condition defines a formal stabilizing transformation Ψ, since then p_{2}, p_{3} will be determined mod J.

The normalization chosen for a transformation p⃗ in Theorem 3.4, which was intended to alter the defining functions, not preserve them, was that z̄p̄_{1} and z̄p_{1} were not in Q when restricted to M̃, and that p_{2}, p_{3} could be arbitrary, but with Q = 0 in the decomposition mod J.

7 An example of an approximate solution

As an example of the approximate normalizing transformation p⃗, consider the following real algebraic surface with a non-degenerate complex tangent in ℂ^{3}, with d = 2k + 1 ≥ 3, and R > 0:

z_{2} = z̄^{2} + \frac{z̄^{d}}{R − z̄},  (17)
z_{3} = zz̄ + \frac{z̄^{d}}{R − z} .  (18)

It is left as an exercise to show that the construction of Theorem 3.4 results in the following normalized p⃗, and as another exercise to verify p⃗ is an exact solution of equations (10) and (11). Note that the functions are holomorphic on Δ_{R}, with weights equal to d − 1, d, d.

p_{1} = \frac{Rz^{2k}}{2(R^{2} − z^{2})} + \frac{z_{2}^{k}}{R} ,
p_{2} = \frac{2z_{3}z^{2k−1}}{R} − \frac{z_{2}^{k+1}}{R^{2} − z_{2}} ,
p_{3} = \frac{2z^{2k+3}z_{2} − R^{2}z^{3}z_{2}^{k} − 2R^{2}z^{2k+3} + R^{4}zz_{2}^{k} + 2R^{4}z^{2k+1} − 2R^{2}z^{2k+1}z_{2}}{2R(R^{2} − z_{2})(R^{2} − z^{2})}

+z_{3} \frac{2zz_{2}^{k+1} − 2R^{2}zz_{2}^{k} + 2Rz_{2}^{k+1} − 2R^{3}z_{2}^{k} + R^{4}z^{2k−1} − R^{2}z^{2k−1}z_{2}}{2R(R^{2} − z_{2})(R^{2} − z^{2})} .

8 Surfaces in higher dimensions

Since the expected real codimension of the complex tangent locus of a real m-manifold in ℂ^{n} is 2(n − m + 1) when m ≤ n, a real 2-manifold M with an isolated complex tangent in ℂ^{n}, n > 3, is quite exceptional. However, for a sufficiently large family of surfaces, one might expect that while most are totally real, some of its members may have an isolated complex tangent. A naturally occuring example of such a family of surfaces in a complex 4manifold is described by [C_{2}], where quadratic rational functions are used to map the real projective plane into ℂP^{4}, and in some special cases, the rational functions define an embedding with exactly one complex tangent.

With coordinate system (z, z_{2}, …, z_{n}) for ℂ^{n}, the complex tangent plane of the surface M can be arranged to be the z-axis as in Section 2, and then the local real analytic defining equations are of the form (z_{2}, …, z_{n}) = (h_{2}(z, z̄), …, h_{n}(z, z̄)), with

h_{i}(z, ζ) = β_{i}zζ + γ_{i}ζ^{2} + θ_{i}ζ^{3}

+α_{i}z^{2} + κ_{i}z^{3} + λ_{i}z^{2}ζ + μ_{i}zζ^{2} + ∑_{a+b≥4} h_{i}^{a,b}z^{a}ζ^{b},

for i = 2, …, n. The non-degeneracy condition is that the (n − 1) × 3 matrix of the β_{i}, γ_{i}, θ_{i} complex coefficients has full rank. In this case, a complex linear transformation of (z_{2}, …, z_{n}) can diagonalize the first three rows of the matrix, and it is clear that α_{i}z^{2} and the remaining cubic terms can be eliminated by weight 2 and 3 transformations of the form z_{i} ↦ z_{i} + p_{i}(z, z_{2}, z_{3}), leaving the following cubic normal form:

z_{2} = z̄^{2} + e_{2}(z, z̄)
z_{3} = zz̄ + e_{3}(z, z̄)
z_{4} = z̄^{3} + e_{4}(z, z̄)
z_{s} = e_{s}(z, z̄),

with degree ≥ 4 functions e_{2}, e_{3}, e_{4}, and e_{s}, 5 ≤ s ≤ n.

Theorem 8.1. Given n ≥ 4, and assuming M is a non-degenerate surface in ℂ^{n}, (in the sense that it has the above defining equations), there exists a local biholomorphism Ψ taking M to the real variety M̃ defined by:

z_{2} = z̄^{2}, z_{3} = zz̄, z_{4} = z̄^{3}, z_{s} = 0.

Proof. As in Theorem 3.4, the main idea is that any series e(z, z̄) can be decomposed as a sum of three series, but the z̄^{2k+1} terms now can be normalized using z_{4} = z̄^{3}. Ψ need not involve a z ↦ z +p_{1} component, and this is such a significant simplification that the rapid convergence argument will not be needed. Rather than constructing an approximate Ψ as in Theorem 3.4, the exact inverse map ψ, defined by

ψ(z, z_{2}, …, z_{n}) = (z, z_{2} + P_{2}(z, z_{2}, z_{3}, z_{4}), …, z_{n} + P_{n}(z, z_{2}, z_{3}, z_{4})),

can be directly computed in terms of e_{2}, …, e_{n}. The condition that defines a normalizing ψ is that if z⃗ ∈ M̃, then ψ(z⃗) ∈ M, so that the following equations must hold when z⃗ =(z, z_{2}, …, z_{n}) is restricted to (z, z̄^{2}, zz̄, z̄^{3}, 0, …, 0):

z_{2} + P_{2}(z⃗) = z̄^{2} + e_{2}(z, z̄)
z_{3} + P_{3}(z⃗) = zz̄ + e_{3}(z, z̄)
z_{4} + P_{4}(z⃗) = z̄^{3} + e_{4}(z, z̄)
z_{s} + P_{s}(z⃗) = e_{s}(z, z̄).

Evidently these equations can be solved exactly, by showing that any real analytic e(z, z̄) is agrees in some neighborhood with the restriction to M̃ of some function P (z, z_{2}, z_{3}, z_{4}) that is complex analytic on a polydisc in ℂ^{n}. To solve for P_{i} in terms of e_{i}, i = 2, …, n, normalize P_{i} so that

P_{i}(z, z_{2}, z_{3}, z_{4}) = P_{i}^{E}(z, z_{2}) + z_{3}P_{i}^{O}(z, z_{2}) + z_{4}P_{i}^{C}(z_{2}).

(As in Theorem 3.4, the superscripts E and O abbreviate even and odd, and C denotes the use of a “cubic” term.) The function

\frac{1}{2}(e_{i}(0, ζ) − e_{i}(0, −ζ)) = ∑ e_{i}^{0,2k+1}ζ^{2k+1}

begins with at least a fifth degree term; let P_{i}^{C}(z_{2}) = ∑ e_{i}^{0,2k+1}z_{2}^{k−1}, so that z̄^{3}P_{i}^{C}(z̄^{2}) = ∑ e_{i}^{0,2k+1}z̄^{2k+1}. If e_{i}(z, ζ) is complex analytic on D_{r}, and |e_{i}|_{r} < ∞, then the series P_{i}^{C} is convergent for z_{2} < r^{2}, and it and its derivatives are bounded (on a smaller disc) by some multiple of |e_{i}|_{r}, as in Corollary 3.5. The rest of the details are omitted, but straightforward modifications of the calculations in Section 3 will find P_{i}^{E} and P_{i}^{O} in terms of e_{i} and P_{i}^{C}, and Lemma 4.3 will show that ψ is invertible on some neighborhood of the origin.

The complexification of M̃ in ℂ^{4} is the complex surface defined by the ideal ⟨z_{2}z_{3} − z_{1}z_{4}, z_{2}^{3} − z_{4}^{2}, z_{1}z_{2}^{2} − z_{3}z_{4}, z_{1}^{2}z_{2} − z_{3}^{2}⟩.

References

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