CR SINGULARITIES OF REAL THREEFOLDS IN ℂ^{4}

ADAM COFFMAN

Abstract. CR singularities of real threefolds in ℂ^{4} are classified by using holomorphic coordinate changes to transform the quadratic part of the real defining equations into one of a list of normal forms. In the non-degenerate case, it is shown that a real analytic manifold near a CR singular point is formally equivalent to a real algebraic model. Some degenerate cases also have this property.

1. Introduction

If a real 3-manifold M is embedded in ℂ^{4}, then for each point x on M there are two possibilities: the tangent 3-plane at x may contain a complex line, so M is said to be “CR singular” at x, or it may not, so M is said to be “totally real” at x. This article will consider the local extrinsic geometry of a real analytically embedded M near a CR singular point, by finding invariants under biholomorphic coordinate changes. The first main result is a classification of quadratic normal forms for the defining equations. The next result concerns the higher-order terms of the normal forms in the non-degenerate case. It will be shown that in this case, M is formally equivalent to a fixed real algebraic variety, in the sense that there are holomorphic coordinate systems in which the defining equations for M agree with the polynomial normal form to an arbitrarily high degree. The final result is a similar formal stability property for some of the degenerate CR singularities.

The analysis of normal forms near CR singular points is part of the program studying the local equivalence problem for real m-submanifolds of ℂ^{n}, as described in the survey paper, [BER]. Normal forms for CR singular real n-manifolds in ℂ^{n}, n ≥ 2, have been considered by [Bishop], [H_{2}], [M], [W], and others, and for real surfaces in ℂ^{n} (m = 2, n ≥ 3) by [H_{1}] and [C_{2}]. A formal normal form for a CR singular real 4-manifold in ℂ^{5} was found by [Beloshapka] and [C_{1}]. The case m = 3, n = 4 to be considered here was motivated in part by [C_{3}], which considers a certain family of maps ℝP^{2} × ℝP^{1} → ℂP^{4}. Most of those maps are totally real embeddings, but some of the exceptional cases exhibit complex tangents.

2. A quadratic classification of CR singularities

Although we could consider real threefolds in any complex 4-manifold, we will only be considering a small neighborhood, so it will be convenient to regard the ambient complex space to be ℂ^{4}, with coordinates (z_{1}, z_{2}, z_{3}, z_{4}). The real and imaginary parts of the coordinate functions are labeled z_{j} = x_{j} + iy_{j}.

2000 Mathematics Subject Classification. 32V40, 32S05.

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

2.1. A general quadratic normal form.

We begin by assuming M is a real analytic three-dimensional submanifold in ℂ^{4} with a complex tangent at some point. By a translation that moves that point to the origin 0, and then a complex linear transformation of ℂ^{4}, the tangent 3plane T = T0M can be assumed to be the (x_{1}, y_{1}, x_{2})-space, which contains the z_{1}-axis. Then there is some neighborhood Δ of the origin in ℂ^{4} so that the defining equations of M in Δ are in the form of a graph over a neighborhood of the origin in T:

(2.1)  y_{2} = H_{2}(z_{1}, z̄_{1}, x_{2})
z_{3} = h_{3}(z_{1}, z̄_{1}, x_{2}),
z_{4} = h_{4}(z_{1}, z̄_{1}, x_{2}),

where H_{2}, h_{3} and h_{4} are real analytic functions defined in a neighborhood of the origin in T, and vanishing to second order at (x_{1}, y_{1}, x_{2}) = (0, 0, 0), with H_{2} realvalued and h_{3}, h_{4} complex-valued. So, they are of the following form:

H_{2}(z_{1}, z̄_{1}, x_{2}) = α_{2}z_{1}^{2} + β_{2}z_{1}z̄_{1} + γ_{2}z̄_{1}^{2} + δ_{2}z_{1}x_{2} + ϵ_{2}z̄_{1}x_{2} + θ_{2}x_{2}^{2} + E_{2}(z_{1}, z̄_{1}, x_{2}),
h_{3}(z_{1}, z̄_{1}, x_{2}) = α_{3}z_{1}^{2} + β_{3}z_{1}z̄_{1} + γ_{3}z̄_{1}^{2} + δ_{3}z_{1}x_{2} + ϵ_{3}z̄_{1}x_{2} + θ_{3}x_{2}^{2} + e_{3}(z_{1}, z̄_{1}, x_{2}),
h_{4}(z_{1}, z̄_{1}, x_{2}) = α_{4}z_{1}^{2} + β_{4}z_{1}z̄_{1} + γ_{4}z̄_{1}^{2} + δ_{4}z_{1}x_{2} + ϵ_{4}z̄_{1}x_{2} + θ_{4}x_{2}^{2} + e_{4}(z_{1}, z̄_{1}, x_{2}),

with E_{2}, e_{3}, e_{4} having terms of degree three or higher. These functions can be expressed as the restriction to {(z, ζ, x) ∈ ℂ^{3} : ζ = z̄, x = x̄} of the three-variable series:

H_{2}(z, ζ, x) = α_{2}z^{2} + β_{2}zζ + γ_{2}ζ^{2} + δ_{2}zx + ϵ_{2}ζx

+θ_{2}x^{2} + ∑_{a+b+c≥3} E_{2}^{a,b,c}z^{a}ζ^{b}x^{c}, (2.2)

h_{3}(z, ζ, x) = α_{3}z^{2} + β_{3}zζ + γ_{3}ζ^{2} + δ_{3}zx + ϵ_{3}ζx

+θ_{3}x^{2} + ∑_{a+b+c≥3} e_{3}^{a,b,c}z^{a}ζ^{b}x^{c},

h_{4}(z, ζ, x) = α_{4}z^{2} + β_{4}zζ + γ_{4}ζ^{2} + δ_{4}zx + ϵ_{4}ζx

+θ_{4}x^{2} + ∑_{a+b+c≥3} e_{4}^{a,b,c}z^{a}ζ^{b}x^{c},

each of which converges on the set {(z, ζ, x) : |z| < R_{1}, |ζ| < R_{1}, |x| < R_{2}} to a complex analytic function, but with γ_{2} = \overline{α_{2}}, ϵ_{2} = \overline{δ_{2}}, β_{2} = \overline{β_{2}}, θ_{2} = \overline{θ_{2}}, and with similar restrictions on the coefficients of E_{2}, so that E_{2}(z_{1}, z̄_{1}, x_{2}) is real-valued.

Definition 2.1. A (formal, with complex coefficient C) monomial of the form Cz^{a}ζ^{b}x^{c} has “degree” a + b + c. A power series (convergent or formal) in three variables e(z, ζ, x) = ∑ e^{a,b,c}z^{a}ζ^{b}x^{c}, is said to have “degree” n if e^{a,b,c} = 0 for all (a, b, c) such that a + b + c < n. Sometimes a series of degree n will be abbreviated O(n). An ordered triple of series (e, f, g) has degree n if all its components have degree n.

Definition 2.2. Similarly for four variables, a monomial of the form Cz_{1}^{a}z_{2}^{b}z_{3}^{c}z_{4}^{d} has degree a + b + c + d, but we will more often work with the “weight,” a + b + 2c + 2d. A series p(z_{1}, z_{2}, z_{3}, z_{4}) = ∑ p^{abcd}z_{1}^{a}z_{2}^{b}z_{3}^{c}z_{4}^{d} has “weight” n if p^{abcd} = 0 when a + b + 2c + 2d < n.

We consider the effect of a coordinate change of the following form:

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

where p_{1}, p_{2}, p_{3}, p_{4} are holomorphic functions with no linear or constant terms (so they have weight 2). Since this transformation of ℂ^{4} has its linear part equal to the identity map, it is invertible on some neighborhood of the origin. In the following calculations, we will neglect considering the size of that neighborhood, and consider only points close enough to the origin. We denote the real and imaginary parts of the new coordinates z̃_{j} = x̃_{j} + ỹ_{j}.

As the first special case of a transformation of the form (2.3) to be used, let p_{1} be identically zero and let p_{2}, p_{3}, p_{4} be homogeneous quadratic polynomials. Given a point on M near 0, its coordinates z⃗ = (z_{1}, …, z_{4}) satisfy Im(z_{2}) − H_{2} = z_{3} − h_{3} = z_{4} − h_{4} = 0. The new coordinates satisfy:

z̃_{3} − (β_{3}z̃_{1} ¯z̃_{1} + γ_{3} ¯z̃_{1}^{2} + ϵ_{3} ¯z̃_{1}Re(z̃_{2})) (2.4)

= z_{3} + p_{3}(z⃗) − (β_{3}z_{1}z̄_{1} + γ_{3}z̄_{1}^{2} + ϵ_{3}z̄_{1}Re(z_{2} + p_{2}(z⃗)))
= α_{3}z_{1}^{2} + δ_{3}z_{1}x_{2} + θ_{3}x_{2}^{2}

+p_{3}(z_{1}, x_{2} + iH_{2}, h_{3}, h_{4}) − ϵ_{3}z̄_{1}Re(p_{2}(z_{1}, x_{2} + iH_{2}, h_{3}, h_{4})) + e_{3}(z_{1}, z̄_{1}, x_{2})

= α_{3}z_{1}^{2} + δ_{3}z_{1}x_{2} + θ_{3}x_{2}^{2} + p_{3}(z_{1}, x_{2}, 0, 0) + O(3)
= α_{3}z̃_{1}^{2} + δ_{3}z̃_{1}x̃_{2} + θ_{3}x̃_{2}^{2} + p_{3}(z̃_{1}, x̃_{2}, 0, 0) + Õ(3),

where O(3) denotes a convergent series of degree three or higher in z_{1}, z̄_{1}, x_{2}, and Õ(3) denotes another series of degree three or higher in z̃_{1}, ¯ z̃_{1}, x̃_{2} (also convergent, possibly on a different neighborhood). The last step of the above process converts back to the z̃ coordinates, using the fact that for points on M sufficiently close to the origin, z_{1} and x_{2} can be expressed as real analytic functions of z̃_{1} and x̃_{2}; the details are given by Lemma 2.3, below.

This calculation shows that the coefficients α_{3}, δ_{3}, θ_{3} can be altered to attain any complex values, by a suitable choice of p_{3} — for example, the transformation z̃_{3} = z_{3} −α_{3}z_{1}^{2} −δ_{3}z_{1}z_{2} −θ_{3}z_{2}^{2} eliminates these terms, so that the defining equations in the z̃ coordinate system will be of the form z̃_{3} = β_{3}z̃_{1} ¯z̃_{1}+γ_{3} ¯z̃_{1}^{2}+ϵ_{3} ¯z̃_{1}x̃_{2}+Õ(3). We note that such a quadratic transformation does not affect the quadratic coefficients of H_{2} or h_{4}, but may change their higher-degree terms.

A similar calculation shows that the coefficients α_{4}, δ_{4}, θ_{4} can be altered arbitrarily by choice of p_{4}, without changing the quadratic terms of H_{2} or h_{3}.

The quantity p_{2} can be used to transform H_{2}:

Im(z̃_{2} − iβ_{2}z̃_{1} ¯z̃_{1}) (2.5)

= Im(z_{2} + p_{2}(z⃗) − iβ_{2}z_{1}z̄_{1})
= H_{2}(z_{1}, z̄_{1}, x_{2}) + Im(p_{2}(z_{1}, x_{2} + iH_{2}, h_{3}, h_{4})) − β_{2}z_{1}z̄_{1}
= α_{2}z_{1}^{2} + \overline{α_{2}}z̄_{1}^{2} + δ_{2}z_{1}x_{2} + \overline{δ_{2}}z̄_{1}x_{2} + θ_{2}x_{2}^{2}
+Im(p_{2}(z_{1}, x_{2}, 0, 0)) + O(3)
= Re(2α_{2}z_{1}^{2} + 2δ_{2}z_{1}x_{2} + θ_{2}x_{2}^{2} − ip_{2}(z_{1}, x_{2}, 0, 0) + O(3))
= Re(2α_{2}z̃_{1}^{2} + 2δ_{2}z̃_{1}x̃_{2} + θ_{2}x̃_{2}^{2} − ip_{2}(z̃_{1}, x̃_{2}, 0, 0) + Õ(3)),

the last step using Lemma 2.3 again. So, the complex coefficients α_{2}, δ_{2} and the real coefficient θ_{2} can be altered arbitrarily by choice of p_{2} — for example, the transformation z̃_{2} = z_{2} − 2iα_{2}z_{1}^{2} − 2iδ_{2}z_{1}z_{2} − iθ_{2}z_{2}^{2} eliminates these terms so that the defining equations in the z̃ coordinate system will be of the form Im(z̃_{2}) = β_{2}z̃_{1} ¯z̃_{1} + Õ(3). Such a quadratic transformation does not change β_{2}, and does not affect the quadratic coefficients of h_{3} or h_{4}, but may change their higher-degree terms.

So, the conclusion is that for any CR singular submanifold M, there exists a quadratic coordinate transformation of the form (2.3) with p_{1} = 0, so that M has the following general normal form. In a local coordinate system (z_{1}, z_{2}, z_{3}, z_{4}) in some neighborhood of the CR singularity, the defining equations of M are of the form (2.1), with

(2.6)  y_{2} = H_{2}(z_{1}, z̄_{1}, x_{2}) = β_{2}z_{1}z̄_{1} + O(3)
z_{3} = h_{3}(z_{1}, z̄_{1}, x_{2}) = β_{3}z_{1}z̄_{1} + γ_{3}z̄_{1}^{2} + ϵ_{3}z̄_{1}x_{2} + O(3)
z_{4} = h_{4}(z_{1}, z̄_{1}, x_{2}) = β_{4}z_{1}z̄_{1} + γ_{4}z̄_{1}^{2} + ϵ_{4}z̄_{1}x_{2} + O(3).

The following Lemma states that for points on M, the coordinates z_{1}, x_{2} can be expressed as real analytic functions of z̃_{1}, x̃_{2} in some neighborhood of (0, 0) in ℂ × ℝ, and that these functions depend on the defining equations of M and the p_{1}, p_{2} components of the coordinate transformation (2.3). We are interested mostly in the existence of the neighborhood and not its size, about which something could be said, given more information about p_{1}, p_{2}, H_{2}, h_{3}, h_{4}, in analogy with the results of [C_{2}].

Lemma 2.3. Given any functions p_{1}(z_{1}, z_{2}, z_{3}, z_{4}), p_{2}(z_{1}, z_{2}, z_{3}, z_{4}) which are holomorphic in a neighborhood of the origin of ℂ^{4} and have weight 2, and given H_{2}, h_{3}, h_{4} as in (2.2) which define M in a neighborhood of the origin in ℂ^{4}, there exist functions ϕ_{1}, ϕ_{3} : ℂ^{3} → ℂ which are holomorphic on some neighborhood of the origin and which vanish to at least second order there, such that for points (z_{1}, z_{2}, z_{3}, z_{4}) on M and sufficiently close to the origin, if z̃_{1} = z_{1} +p_{1}(z_{1}, z_{2}, z_{3}, z_{4}) and z̃_{2} = z_{2} + p_{2}(z_{1}, z_{2}, z_{3}, z_{4}), then

z_{1} = z̃_{1} + ϕ_{1}(z̃_{1}, ¯ z̃_{1}, Re(z̃_{2}))
Re(z_{2}) = Re(z̃_{2}) + ϕ_{3}(z̃_{1}, ¯ z̃_{1}, Re(z̃_{2})).

Proof. Consider the map τ that takes (z, ζ, x) to the ordered triple:

(z + p_{1}(z, x + iH_{2}(z, ζ, x), h_{3}(z, ζ, x), h_{4}(z, ζ, x)),

ζ + \overline{p_{1}(ζ̄, x̄ + iH_{2}(ζ̄, z̄, x̄), h_{3}(ζ̄, z̄, x̄), h_{4}(ζ̄, z̄, x̄))},

x + \frac{1}{2}p_{2}(z, x + iH_{2}(z, ζ, x), h_{3}(z, ζ, x), h_{4}(z, ζ, x))

+\frac{1}{2}\overline{p_{2}(ζ̄, x̄ + iH_{2}(ζ̄, z̄, x̄), h_{3}(ζ̄, z̄, x̄), h_{4}(ζ̄, z̄, x̄))}).

Note that τ is holomorphic and invertible in a neighborhood of the origin of ℂ^{3}, with identity linear part. Its inverse is given by a function ϕ such that τ(ϕ(z′, ζ′, x′)) = (z′, ζ′, x′), with

ϕ(z′, ζ′, x′) = (z′ + ϕ_{1}(z′, ζ′, x′), ζ′ + ϕ_{2}(z′, ζ′, x′), x′ + ϕ_{3}(z′, ζ′, x′)).

Also note that if τ(z, ζ, x) = (z′, ζ′, x′), then τ(ζ̄, z̄, x̄) = (ζ̄′, z̄′, x̄′). Denoting by C the antiholomorphic involution of ℂ^{3} where C(z, ζ, x) = (ζ̄, z̄, x̄), it follows from C ∘ τ ∘ C = τ that C ∘ ϕ ∘ C = ϕ on some neighborhood of the origin of ℂ^{3}.

By construction, for x_{2} = Re(z_{2}),

τ(z_{1}, z̄_{1}, x_{2}) = (z_{1} + p_{1}(z_{1}, x_{2} + iH_{2}(z_{1}, z̄_{1}, x_{2}), h_{3}(z_{1}, z̄_{1}, x_{2}), h_{4}(z_{1}, z̄_{1}, x_{2})),
\overline{z_{1} + p_{1}(z_{1}, x_{2} + iH_{2}(z_{1}, z̄_{1}, x_{2}), h_{3}(z_{1}, z̄_{1}, x_{2}), h_{4}(z_{1}, z̄_{1}, x_{2}))},

x_{2} + Re(p_{2}(z_{1}, x_{2} + iH_{2}(z_{1}, z̄_{1}, x_{2}), h_{3}(z_{1}, z̄_{1}, x_{2}), h_{4}(z_{1}, z̄_{1}, x_{2})))).

If a point (z_{1}, z_{2}, z_{3}, z_{4}) is on M and close enough to the origin, then it is of the form (z_{1}, x_{2} + iH_{2}(z_{1}, z̄_{1}, x_{2}), h_{3}(z_{1}, z̄_{1}, x_{2}), h_{4}(z_{1}, z̄_{1}, x_{2})), and

z̃_{1} = z_{1} + p_{1}(z_{1}, z_{2}, z_{3}, z_{4})
= z_{1} + p_{1}(z_{1}, x_{2} + iH_{2}(z_{1}, z̄_{1}, x_{2}), h_{3}(z_{1}, z̄_{1}, x_{2}), h_{4}(z_{1}, z̄_{1}, x_{2})),
z̃_{2} = z_{2} + p_{2}(z_{1}, z_{2}, z_{3}, z_{4})
= z_{2} + p_{2}(z_{1}, x_{2} + iH_{2}(z_{1}, z̄_{1}, x_{2}), h_{3}(z_{1}, z̄_{1}, x_{2}), h_{4}(z_{1}, z̄_{1}, x_{2})),

so τ(z_{1}, z̄_{1}, x_{2}) = (z̃_{1}, ¯ z̃_{1}, Re(z̃_{2})) = τ(ϕ(z̃_{1}, ¯ z̃_{1}, Re(z̃_{2}))). The conclusion is that

z_{1} = z̃_{1} + ϕ_{1}(z̃_{1}, ¯ z̃_{1}, Re(z̃_{2}))
z̄_{1} = ¯ z̃_{1} + ϕ_{2}(z̃_{1}, ¯ z̃_{1}, Re(z̃_{2})) = ¯ z̃_{1} + \overline{ϕ_{1}(z̃_{1}, ¯ z̃_{1}, Re(z̃_{2}))}
x_{2} = Re(z̃_{2}) + ϕ_{3}(z̃_{1}, ¯ z̃_{1}, Re(z̃_{2})) = Re(z̃_{2}) + \overline{ϕ_{3}(z̃_{1}, ¯ z̃_{1}, Re(z̃_{2}))}.

2.2. The non-degenerate case.

Next, we consider some linear transformations of ℂ^{4}, but only those which fix the tangent 3-plane T, so they are of the form

(2.7)  z̃_{4×1} = ( \begin{matrix}c_{1} & ∗ & ∗ & ∗ \\ 0 & r_{2} & ∗ & ∗ \\ 0 & 0 & c_{3} & c_{4} \\ 0 & 0 & c_{5} & c_{6} \end{matrix} ) z⃗_{4×1},

with complex entries such that c_{1}r_{2} ≠ 0, r_{2} = \overline{r_{2}}, c_{3}c_{6} − c_{4}c_{5} ≠ 0.

First, consider the linear coordinate change z̃_{3} = c_{3}z_{3} + c_{4}z_{4}, z̃_{4} = c_{5}z_{3} + c_{6}z_{6}. In this coordinate system, the coefficients of H_{2} are not changed, and we get

z̃_{3} = c_{3}(β_{3}z_{1}z̄_{1} + γ_{3}z̄_{1}^{2} + ϵ_{3}z̄_{1}x_{2} + O(3)) + c_{4}(β_{4}z_{1}z̄_{1} + γ_{4}z̄_{1}^{2} + ϵ_{4}z̄_{1}x_{2} + O(3))
= (c_{3}β_{3} + c_{4}β_{4})z̃_{1} ¯z̃_{1} + (c_{3}γ_{3} + c_{4}γ_{4})¯z̃_{1}^{2} + (c_{3}ϵ_{3} + c_{4}ϵ_{4})¯z̃_{1}x̃_{2} + Õ(3),

and similarly

z̃_{4} = (c_{5}β_{3} + c_{6}β_{4})z̃_{1} ¯z̃_{1} + (c_{5}γ_{3} + c_{6}γ_{4})¯z̃_{1}^{2} + (c_{5}ϵ_{3} + c_{6}ϵ_{4})¯z̃_{1}x̃_{2} + Õ(3).

At this point we introduce the “first non-degeneracy condition,” which is satisfied if:

(2.8)  det ( \begin{matrix}β_{3} & γ_{3} \\ β_{4} & γ_{4} \end{matrix} ) ≠ 0,

so that there is a complex linear coordinate change transforming the general normal form (2.6) to:

(2.9)  y_{2} = β_{2}z_{1}z̄_{1} + E_{2}(z_{1}, z̄_{1}, x_{2})
z_{3} = z̄_{1}^{2} + ϵ_{3}z̄_{1}x_{2} + e_{3}(z_{1}, z̄_{1}, x_{2})
z_{4} = z_{1}z̄_{1} + ϵ_{4}z̄_{1}x_{2} + e_{4}(z_{1}, z̄_{1}, x_{2}).

Next, consider a linear transformation of the form z̃_{2} = z_{2} + irz_{4}, for some real r, so that for (z_{1}, z_{2}, z_{3}, z_{4}) on M:

Im(z̃_{2}) (2.10)

= Im(z_{2} + irz_{4})
= H_{2}(z_{1}, z̄_{1}, x_{2}) + Im(irh_{4}(z_{1}, z̄_{1}, x_{2}))
= β_{2}z_{1}z̄_{1} + E_{3}(z_{1}, z̄_{1}, x_{2}) + rz_{1}z̄_{1} + Re(rϵ_{4}z̄_{1}x_{2}) + Re(re_{4}(z_{1}, z̄_{1}, x_{2}))
= (β_{2} + r)z_{1}z̄_{1} + \frac{1}{2}rϵ_{4}z̄_{1}x_{2} + \frac{1}{2}r\overline{ϵ_{4}}z_{1}x_{2} + O(3)
= (β_{2} + r)z̃_{1} ¯z̃_{1} + \frac{1}{2}rϵ_{4} ¯z̃_{1}x̃_{2} + \frac{1}{2}r\overline{ϵ_{4}}z̃_{1}x̃_{2} + Õ(3).

Even though z̃_{2} = z_{2} + irz_{4} is a linear transformation, the hypotheses of Lemma 2.3 are satisfied, since in this case p_{2}(z⃗) = irz_{4} has weight 2, so τ(z, ζ, x) is defined by:

(z, ζ, x + \frac{1}{2}(irzζ + irϵ_{4}ζx + ire_{4}(z, ζ, x)) + \frac{1}{2}\overline{(irζ̄z̄ + irϵ_{4}z̄x̄ + ire_{4}(ζ̄, z̄, x̄))}),

which has identity linear part. By inspection of (2.4) with p_{2}(z⃗) = irz_{4}, the transformation does not affect the quadratic coefficients of h_{3} or h_{4} but may change their higher-degree terms. Clearly, choosing r = −β_{2} will eliminate the z̃_{1} ¯z̃_{1} term, leaving only the ¯ z̃_{1}x̃_{2} and z̃_{1}x̃_{2} terms which can be eliminated by another quadratic transformation as in (2.5) without re-introducing a z̃_{1} ¯z̃_{1} term.

Another linear transformation is z̃_{1} = z_{1} + cz_{2}, for some complex coefficient c.

z̃_{4} − z̃_{1} ¯z̃_{1} = z_{4} − (z_{1} + cz_{2})(z̄_{1} + c̄z̄_{2})
= ϵ_{4}z̄_{1}x_{2} + e_{4}(z_{1}, z̄_{1}, x_{2})
−cz̄_{1}(x_{2} + iH_{2}(z_{1}, z̄_{1}, x_{2})) − c̄z_{1}(x_{2} − i\overline{H_{2}(z_{1}, z̄_{1}, x_{2})})
−cc̄(x_{2} + iH_{2}(z_{1}, z̄_{1}, x_{2}))(x_{2} − i\overline{H_{2}(z_{1}, z̄_{1}, x_{2})})
= (ϵ_{4} − c)z̄_{1}x_{2} − c̄z_{1}x_{2} − cc̄x_{2}^{2} + O(3).

This time, the hypotheses of Lemma 2.3 are not satisfied, since in this case p_{1}(z⃗) = cz_{2} has weight 1. The map τ(z, ζ, x) defined by

(z + c(x + iH_{2}(z, ζ, x)), ζ + \overline{c(x̄ + iH_{2}(ζ̄, z̄, x̄))}, x)

has an invertible linear part which is not the identity, but the inverse transformation is holomorphic on some neighborhood, and given by

ϕ(z′, ζ′, x′) = (z′ − cx′ + ϕ_{1}, ζ′ − c̄x′ + ϕ_{2}, x′ + ϕ_{3}).

The conclusion analogous to Lemma 2.3 is then that z_{1} = z̃_{1} − cx̃_{2} + ϕ_{1}(z̃_{1}, ¯ z̃_{1}, x̃_{2}), x_{2} = x̃_{2} + ϕ_{3}(z̃_{1}, ¯ z̃_{1}, x̃_{2}) is real analytic, so the new z̃_{4} equation is

z̃_{4} = z̃_{1} ¯z̃_{1} + (ϵ_{4} − c)(¯z̃_{1} − c̄x̃_{2})x̃_{2} − c̄(z̃_{1} − cx̃_{2})x̃_{2} − cc̄x̃_{2}^{2} + Õ(3)
(2.11)  = z̃_{1} ¯z̃_{1} + (ϵ_{4} − c)¯z̃_{1}x̃_{2} − c̄z̃_{1}x̃_{2} + (−ϵ_{4}c̄ + cc̄)x̃_{2}^{2} + Õ(3).

This transformation does not introduce any quadratic terms in H_{2}, but it does change h_{3}:

z̃_{3} − ¯ z̃_{1}^{2} = z_{3} − (z̄_{1} + c̄z̄_{2})^{2}
= ϵ_{3}z̄_{1}x_{2} + e_{3}(z_{1}, z̄_{1}, x_{2})
−2c̄z̄_{1}(x_{2} − i\overline{H_{2}(z_{1}, z̄_{1}, x_{2})}) − c̄^{2}(x_{2} − i\overline{H_{2}(z_{1}, z̄_{1}, x_{2})})^{2}
= (ϵ_{3} − 2c̄)z̄_{1}x_{2} − c̄^{2}x_{2}^{2} + O(3)
= (ϵ_{3} − 2c̄)(¯z̃_{1} − c̄x̃_{2})x̃_{2} − c̄^{2}x̃_{2}^{2} + Õ(3)
(2.12)  = (ϵ_{3} − 2c̄)¯z̃_{1}x̃_{2} + (−ϵ_{3}c̄ + c̄^{2})x̃_{2}^{2} + Õ(3).

From (2.11) and (2.12), we see c can be chosen to eliminate the ¯ z̃_{1}x̃_{2} term from either the h_{3} or the h_{4} series, but it is unlikely that the transformation will simultaneously eliminate the ¯ z̃_{1}x̃_{2} term from both. At this point, we choose to eliminate ¯ z̃_{1}x̃_{2} term from h_{3} by selecting c = \overline{ϵ_{3}}/2, and then another transformation as in (2.4) with suitable p_{3}, p_{4} will clean up the quadratic terms introduced above, giving the following quadratic normal form for any M satisfying the first non-degeneracy condition:

(2.13)  y_{2} = E_{2}(z_{1}, z̄_{1}, x_{2}) = O(3)
z_{3} = z̄_{1}^{2} + e_{3}(z_{1}, z̄_{1}, x_{2})
z_{4} = (z_{1} + ϵ_{4}x_{2})z̄_{1} + e_{4}(z_{1}, z̄_{1}, x_{2}).

The “second non-degeneracy condition” is that ϵ_{4} ≠ 0, after M is put into the above normal form (so the first non-degeneracy condition is assumed). In this case, an invertible linear coordinate change of the form z̃_{1} = c_{1}z_{1}, z̃_{3} = c_{3}z_{3}, z̃_{4} = c_{6}z_{4} gives:

(2.14)  ỹ_{2} = E_{2}(\frac{z̃_{1}}{c_{1}} , \frac{¯z̃_{1}}{\overline{c_{1}}} , x̃_{2}) = Õ(3)
z̃_{3} = c_{3}((\frac{¯z̃_{1}}{\overline{c_{1}}} )^{2} + e_{3}(\frac{z̃_{1}}{c_{1}} , \frac{¯z̃_{1}}{\overline{c_{1}}} , x̃_{2}))
z̃_{4} = c_{6}(\frac{z̃_{1}}{c_{1}} \frac{¯z̃_{1}}{\overline{c_{1}}} + ϵ_{4} \frac{¯z̃_{1}}{\overline{c_{1}}} x̃_{2} + e_{4}(\frac{z̃_{1}}{c_{1}} , \frac{¯z̃_{1}}{\overline{c_{1}}} , x̃_{2})),

and choosing (c_{1}, c_{3}, c_{6}) = (1/ϵ_{4}, 1/\overline{ϵ_{4}}^{2}, 1/|ϵ_{4}|^{2}) normalizes the coefficient of ¯ z̃_{1}x̃_{2} in h_{4} to 1.

2.3. Classifying the degenerate cases. Returning to (2.6), suppose that the determinant from (2.8) is zero. Then a linear transformation (2.7) could put the

rank 1 coefficient matrix ( \begin{matrix}β_{3} & γ_{3} \\ β_{4} & γ_{4} \end{matrix} ) into one of three normal forms, taking the

defining equations (2.6) to one of the following cases:

(2.15)  y_{2} = β_{2}z_{1}z̄_{1} + E_{2}(z_{1}, z̄_{1}, x_{2})
z_{3} = z_{1}z̄_{1} + γ_{3}z̄_{1}^{2} + ϵ_{3}z̄_{1}x_{2} + e_{3}(z_{1}, z̄_{1}, x_{2})
z_{4} = ϵ_{4}z̄_{1}x_{2} + e_{4}(z_{1}, z̄_{1}, x_{2}).
(2.16)  y_{2} = β_{2}z_{1}z̄_{1} + E_{2}(z_{1}, z̄_{1}, x_{2})
z_{3} = z̄_{1}^{2} + ϵ_{3}z̄_{1}x_{2} + e_{3}(z_{1}, z̄_{1}, x_{2})
z_{4} = ϵ_{4}z̄_{1}x_{2} + e_{4}(z_{1}, z̄_{1}, x_{2}).

(2.17)  y_{2} = β_{2}z_{1}z̄_{1} + E_{2}(z_{1}, z̄_{1}, x_{2})
z_{3} = ϵ_{3}z̄_{1}x_{2} + e_{3}(z_{1}, z̄_{1}, x_{2})
z_{4} = ϵ_{4}z̄_{1}x_{2} + e_{4}(z_{1}, z̄_{1}, x_{2}).

In (2.15), a re-scaling as in (2.14) can transform the coefficient γ_{3} to a non-negative real number, resembling Bishop’s invariant of surfaces in ℂ^{2}, ([Bishop]). The z_{3} expression in (2.16) could be considered the “γ_{3} = ∞” case of (2.15). If ϵ_{4} ≠ 0, another linear transformation of the z_{3}, z_{4} coordinates could re-scale ϵ_{4} = 1 and eliminate ϵ_{3} from (2.15), (2.16), or (2.17). Otherwise, if ϵ_{4} = 0, ϵ_{3} could be eliminated from (2.15) or (2.16) by a linear transformation of the form z̃_{1} = z_{1} +cz_{2} as in (2.12), unless γ_{3} = \frac{1}{2} in (2.15), where only the real part of ϵ_{3} can be eliminated, and Im(ϵ_{3}) can be re-scaled to 1 or 0. The γ_{3} = \frac{1}{2}, ϵ_{3} = i, ϵ_{4} = 0 case (type (IVp) in the table below) corresponds to the parabolic normal form of [W]. In (2.17), if ϵ_{4} = 0, ϵ_{3} could be re-scaled to be either 1 or 0. In (2.15), a linear transformation of the form z̃_{2} = z_{2} + irz_{3} as in (2.10) will eliminate β_{2}. In the other cases, β_{2} can only be re-scaled to be either 1 or 0.

To summarize, for any real threefold M in ℂ^{4} with a CR singular point, there is some coordinate system in a neighborhood of that point so that the defining equations of M in that neighborhood are of the form

y_{2} = Q_{2} + E_{2}(z_{1}, z̄_{1}, x_{2})
z_{3} = q_{3} + e_{3}(z_{1}, z̄_{1}, x_{2})
z_{4} = q_{4} + e_{4}(z_{1}, z̄_{1}, x_{2}),

where E_{2}, e_{3}, e_{4} vanish to third order, E_{2} is real-valued, and Q_{2}, q_{3}, q_{4} are quadratic quantities falling into exactly one form from the following list.

(I)  Q_{2} = 0  (V)  Q_{2} = z_{1}z̄_{1} (IX) Q_{2} = z_{1}z̄_{1}
q_{3} = z̄_{1}^{2}  q_{3} = z̄_{1}^{2}  q_{3} = 0
q_{4} = z_{1}z̄_{1} + x_{2}z̄_{1}  q_{4} = z̄_{1}x_{2}  q_{4} = z̄_{1}x_{2}
(II) Q_{2} = 0  (VI) Q_{2} = z_{1}z̄_{1} (X) Q_{2} = z_{1}z̄_{1}
q_{3} = z̄_{1}^{2}  q_{3} = z̄_{1}^{2}  q_{3} = 0
q_{4} = z_{1}z̄_{1}  q_{4} = 0  q_{4} = 0
(III) Q_{2} = 0  (VII) Q_{2} = 0  (XI) Q_{2} = 0
q_{3} = z_{1}z̄_{1} + γz̄_{1}^{2}, γ ≥ 0  q_{3} = z̄_{1}^{2}  q_{3} = 0
q_{4} = z̄_{1}x_{2}  q_{4} = z̄_{1}x_{2}  q_{4} = z̄_{1}x_{2}
(IV) Q_{2} = 0  (VIII) Q_{2} = 0  (XII) Q_{2} = 0
q_{3} = z_{1}z̄_{1} + γz̄_{1}^{2}, γ ≥ 0  q_{3} = z̄_{1}^{2}  q_{3} = 0
q_{4} = 0  q_{4} = 0  q_{4} = 0
(IVp) Q_{2} = 0
q_{3} = z_{1}z̄_{1} + \frac{1}{2}z̄_{1}^{2} + iz̄_{1}x_{2}
q_{4} = 0

Type (I) is the non-degenerate quadratic normal form, (2.13) with ϵ_{4} scaled to 1 as in (2.14). Type (II) is the quadratic normal form for a manifold satisfying the first non-degeneracy condition but not the second, so ϵ_{4} = 0 in (2.13). The middle column has the γ = ∞ cases.

2.4. Examples. Most real analytic threefolds in ℂ^{4} are totally real at every point — a CR singularity is topologically unstable, in the sense that a real manifold with a CR singular point can be perturbed by a small amount to become totally real. However, simple examples of CR singular submanifolds in complex Euclidean 4-space are not hard to construct and we consider a few here.

Example 2.4. Singularities of type (I), (III) with γ > \frac{1}{2}, (VII), and (XI) can occur in projections of the real Segre threefold from ℂP^{5} to ℂP^{4}, as shown in [C_{3}].

Example 2.5. Among the three-dimensional real affine subspaces of ℂ^{4}, most are totally real. The rest, those that contain a complex line, are related to the z_{1}, x_{2}- subspace by some complex affine transformation, and have a type (XII) singularity at every point.

Example 2.6. Considering the Euclidean 3-sphere contained inside some fourdimensional real affine subspace of ℂ^{4}, again such a subspace in general position is totally real, and so is the sphere it contains. One exceptional case is that the real affine subspace is contained in a complex 3-subspace but is not itself a complex subspace. In this case, the real subspace necessarily contains a complex line, and the standard hypersphere is CR singular along a real circle. A complex affine coordinate change takes a such a sphere to {x_{1}^{2} + y_{1}^{2} + x_{2}^{2} + (x_{3} − 1)^{2} = 1} inside the z_{1}, x_{2}, x_{3}-subspace, and the CR singular locus is the circle of points on the sphere with z_{1} = 0. In a neighborhood of the origin, the defining equations of the sphere become y_{2} = y_{3} = z_{4} = 0 and

x_{3} = 1 − \sqrt{1 − x_{1}^{2} − y_{1}^{2} − x_{2}^{2}}
⟹ z_{3} = \frac{1}{2}(z_{1}z̄_{1} + x_{2}^{2}) + O(4).

This is a type (IV) singularity with γ = 0. More generally, the quadric {ax_{1}^{2} + y_{1}^{2} + x_{2}^{2} + (x_{3} − 1)^{2} = 1\}, for a ∈ ℝ, has a CR singularity of type (IV) with γ = |\frac{a−1}{2(a+1)} | for a ≠ −1, or type (VIII) for a = −1. So, each γ in the range [0, ∞] has a representative of this form, and it is exactly the well-known Bishop invariant for n-manifolds in ℂ^{n}, with n = 3.

Example 2.7. The other exceptional case of a four-dimensional real affine subspace of ℂ^{4} is that it is a two-dimensional complex affine subspace. Then, a standard 3-sphere that it contains is a real hypersurface in a complex 2-manifold, so it has a complex tangent at every point, and is CR singular at every point when considered as a real submanifold of ℂ^{4}. A complex affine coordinate change takes such a sphere to {x_{1}^{2} + y_{1}^{2} + x_{2}^{2} + (y_{2} − 1)^{2} = 1} in the z_{1}, z_{2}-subspace, and in a neighborhood of the origin, the defining equations of the sphere become z_{3} = z_{4} = 0 and

y_{2} = 1 − \sqrt{1 − x_{1}^{2} − y_{1}^{2} − x_{2}^{2}}
= \frac{1}{2}(z_{1}z̄_{1} + x_{2}^{2}) + O(4).

This is a type (X) singularity. Again generalizing to the quadric {ax_{1}^{2} + y_{1}^{2} + x_{2}^{2} + (y_{2} − 1)^{2} = 1}, for a ∈ ℝ, the CR singularity is of type (X) for all a except a = −1, where the z_{1}z̄_{1} term has coefficient 0 and the quadratic normal form is of type

3. Higher-order terms in the non-degenerate case

After a transformation as in the previous Section, the equations near a nondegenerate (type (I)) CR singularity are in the following form:

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

where E_{2}, e_{3}, e_{4} have degree 3 in z_{1}, z̄_{1}, x_{2}, and are real analytic, converging for z_{1}, x_{2} in some neighborhood of the origin in the z_{1}, x_{2} 3-space. It will be shown in this Section that there exists p⃗ = (p_{1}, p_{2}, p_{3}, p_{4}) defining a coordinate transformation as in (2.3) such that the defining equations in the new coordinates are as in (3.1), but with E_{2}, e_{3}, e_{4} vanishing to any degree n, n ≥ 3.

We will consider formal coordinate transformations p⃗ as in (2.3) where p_{1} has weight 2 and p_{2}, p_{3}, p_{4} have weight 3. We will think of these transformations as having identity linear part, although a weight 2 transformation of z_{1} could have linear terms of the form z̃_{1} = z_{1} + c_{3}z_{3} + c_{4}z_{4}. The effect of the transformation is that for points on M, of the form z⃗ = (z_{1}, x_{2} + iE_{2}, z̄_{1}^{2} + e_{3}, (z_{1} + x_{2})z̄_{1} + e_{4}),

(3.2)  Im(z̃_{2}) = Im(z_{2} + p_{2}(z⃗))
= Im(x_{2} + iE_{2}(z_{1}, z̄_{1}, x_{2}) + p_{2}(z⃗))
= E_{2}(z_{1}, z̄_{1}, x_{2}) + Im(p_{2}),
(3.3)  z̃_{3} − ¯ z̃_{1}^{2} = z_{3} + p_{3}(z⃗) − \overline{(z_{1} + p_{1}(z⃗))}^{2}
= e_{3} + p_{3} − 2z̄_{1}\overline{p_{1}} − \overline{p_{1}}^{2},

z̃_{4} − (z̃_{1} + x̃_{2})¯z̃_{1} (3.4)

= z_{4} + p_{4}(z⃗) − (z_{1} + p_{1}(z⃗) + x_{2} + Re(p_{2}(z⃗)))\overline{(z_{1} + p_{1}(z⃗))}
= e_{4} + p_{4} − (z_{1} + x_{2})\overline{p_{1}} − z̄_{1}(p_{1} + Re(p_{2})) − p_{1}\overline{p_{1}} − Re(p_{2})\overline{p_{1}}.

The “normal form” problem is then the following: given convergent series E_{2}, e_{3}, e_{4} as in (3.1), with degree ≥ 3, find series p_{1}, p_{2}, p_{3}, p_{4} as in (2.3) which are solutions of these non-linear equations:

(3.5) 0 = E_{2}(z_{1}, z̄_{1}, x_{2}) + Im(p_{2}(z_{1}, x_{2} + iE_{2}, z̄_{1}^{2} + e_{3}, (z_{1} + x_{2})z̄_{1} + e_{4}))
0 = e_{3}(z_{1}, z̄_{1}, x_{2}) + p_{3}(z_{1}, x_{2} + iE_{2}, z̄_{1}^{2} + e_{3}, (z_{1} + x_{2})z̄_{1} + e_{4})
−2z̄_{1}\overline{p_{1}(z_{1}, x_{2} + iE_{2}, z̄_{1}^{2} + e_{3}, (z_{1} + x_{2})z̄_{1} + e_{4})}
−\overline{p_{1}(z_{1}, x_{2} + iE_{2}, z̄_{1}^{2} + e_{3}, (z_{1} + x_{2})z̄_{1} + e_{4})}^{2}
0 = e_{4}(z_{1}, z̄_{1}, x_{2}) + p_{4}(z_{1}, x_{2} + iE_{2}, z̄_{1}^{2} + e_{3}, (z_{1} + x_{2})z̄_{1} + e_{4})
−(z_{1} + x_{2})\overline{p_{1}(z_{1}, x_{2} + iE_{2}, z̄_{1}^{2} + e_{3}, (z_{1} + x_{2})z̄_{1} + e_{4})}
−z̄_{1}p_{1}(z_{1}, x_{2} + iE_{2}, z̄_{1}^{2} + e_{3}, (z_{1} + x_{2})z̄_{1} + e_{4})
−z̄_{1}Re(p_{2}(z_{1}, x_{2} + iE_{2}, z̄_{1}^{2} + e_{3}, (z_{1} + x_{2})z̄_{1} + e_{4}))
−|p_{1}(z_{1}, x_{2} + iE_{2}, z̄_{1}^{2} + e_{3}, (z_{1} + x_{2})z̄_{1} + e_{4})|^{2} − Re(p_{2}(z⃗))\overline{p_{1}(z⃗)}.

If, given E_{2}, e_{3}, e_{4}, we can find series solutions p_{1}, p_{2}, p_{3}, p_{4} of the above equations, which converge on some neighborhood of 0 ∈ ℂ^{4}, then M is “analytically equivalent” to the “polynomial model” {ỹ_{2} = 0, z̃_{3} = ¯ z̃_{1}^{2}, z̃_{4} = (z̃_{1} + x̃_{2})¯z̃_{1}}. More generally, if there exists any formal series solution, then we can say M is “formally equivalent” to the polynomial model.

Theorem 3.1. Given M with the non-degenerate quadratic normal form (type (I), as in (3.1)), there exist formal series p_{1}, p_{2}, p_{3}, p_{4} which are solutions of the system of equations (3.5), so that M is formally equivalent to the polynomial model.

Proof. The idea is to iterate the following step: given series e⃗ = (E_{2}, e_{3}, e_{4}) of degree n ≥ 3, we will find a transformation (2.3) using series p⃗ = (p_{1}, p_{2}, p_{3}, p_{4}), called an “approximate solution,” so that the defining expressions in the new coordinates have degree ≥ n + 1. The existence of a formal transformation follows, since its terms up to any degree can be determined by composing sufficiently many of the approximate solutions.

We begin by replacing (3.5) by a related system of linear equations:

(3.6) 0 = E_{2}(z_{1}, z̄_{1}, x_{2}) + Im(p_{2}(z_{1}, x_{2}, z̄_{1}^{2}, (z_{1} + x_{2})z̄_{1}))
0 = e_{3}(z_{1}, z̄_{1}, x_{2}) + p_{3}(z_{1}, x_{2}, z̄_{1}^{2}, (z_{1} + x_{2})z̄_{1})
−2z̄_{1}\overline{p_{1}(z_{1}, x_{2}, z̄_{1}^{2}, (z_{1} + x_{2})z̄_{1})}
0 = e_{4}(z_{1}, z̄_{1}, x_{2}) + p_{4}(z_{1}, x_{2}, z̄_{1}^{2}, (z_{1} + x_{2})z̄_{1})
−(z_{1} + x_{2})\overline{p_{1}(z_{1}, x_{2}, z̄_{1}^{2}, (z_{1} + x_{2})z̄_{1})}
−z̄_{1}(p_{1}(z_{1}, x_{2}, z̄_{1}^{2}, (z_{1} + x_{2})z̄_{1}) + Re(p_{2}(z_{1}, x_{2}, z̄_{1}^{2}, (z_{1} + x_{2})z̄_{1}))).

By comparing (3.5) with (3.6), we see that if e⃗ has degree n, and p⃗ is an exact solution of (3.6), with the weight of p_{1} equal to n − 1 and the weight of p_{2}, p_{3}, and p_{4} equal to n, then evaluating the RHS of (3.5) with this p⃗ will result in an expression with degree 2n − 2 (which is ≥ n + 1 for n ≥ 3). So, a solution of (3.6) is an approximate solution of (3.5). The degree 2n − 2 quantity, if converted back to the z̃ coordinates using Lemma 2.3, will still have degree ≥ 2n−2, and, returning to Equations (3.2), (3.3), (3.4), will be the higher-order part of the defining equations in the new coordinate system.

The remainder of the Proof will be the construction of such an approximate solution, assuming n ≥ 3.

We start with the first equation from the system (3.6):

(3.7)  0 = E_{2}(z_{1}, z̄_{1}, x_{2}) + Im(p_{2}(z_{1}, x_{2}, z̄_{1}^{2}, (z_{1} + x_{2})z̄_{1})).

The real-valued real analytic function E_{2} can be written in the following form:

E_{2}(z_{1}, z̄_{1}, x_{2}) = ∑ e_{2}^{abc}z_{1}^{a}z̄_{1}^{b}x_{2}^{c}
= e_{2A} + \overline{e_{2A}} + e_{2B} + e_{2C} + \overline{e_{2C}} + e_{2D} + \overline{e_{2D}} + e_{2E},

12 A. COFFMAN

where E_{2} has degree n (as in Definition 2.1), e_{2}^{bac} = \overline{e_{2}^{abc}} and

e_{2A} = ∑_{a > b, b even} e_{2}^{abc}z_{1}^{a}z̄_{1}^{b}x_{2}^{c}
e_{2B} = ∑_{a even} e_{2}^{aac}z_{1}^{a}z̄_{1}^{a}x_{2}^{c}
e_{2C} = ∑_{a > b, a even, b odd} e_{2}^{abc}z_{1}^{a}z̄_{1}^{b}x_{2}^{c}
e_{2D} = ∑_{a > b, a, b odd} e_{2}^{abc}z_{1}^{a}z̄_{1}^{b}x_{2}^{c}
e_{2E} = ∑_{a odd} e_{2}^{aac}z_{1}^{a}z̄_{1}^{a}x_{2}^{c}.

We further rearrange e_{2D} and e_{2E}, to get:

e_{2D} = (1 + \frac{x_{2}}{z_{1}} )e_{2D} − \frac{x_{2}}{z_{1}} e_{2D} = e_{2F} + e_{2G},
e_{2F} = ∑_{a > b, a, b odd} e_{2}^{abc}z_{1}^{a−1}(z_{1} + x_{2})z̄_{1}z̄_{1}^{b−1}x_{2}^{c}
e_{2G} = − ∑_{a > b, a, b odd} e_{2}^{abc}z_{1}^{a−1}z̄_{1}^{b}x_{2}^{c+1},
e_{2E} = (\frac{1}{2} + \frac{x_{2}}{2z_{1}} + \frac{1}{2} + \frac{x_{2}}{2z̄_{1}} )e_{2E} − (\frac{x_{2}}{2z_{1}} + \frac{x_{2}}{2z̄_{1}} )e_{2E} = e_{2H} + \overline{e_{2H}} + e_{2I} + \overline{e_{2I}},
e_{2H} = \frac{1}{2} ∑_{a odd} e_{2}^{aac}z_{1}^{a−1}z̄_{1}^{a−1}(z_{1} + x_{2})z̄_{1}x_{2}^{c},
e_{2I} = −\frac{1}{2} ∑_{a odd} e_{2}^{aac}z_{1}^{a−1}z̄_{1}^{a}x_{2}^{c+1},

so E_{2} = e_{2A} + \overline{e_{2A}} + e_{2B} + e_{2C} + \overline{e_{2C}} + e_{2F} + \overline{e_{2F}} + e_{2G} + \overline{e_{2G}} + e_{2H} + \overline{e_{2H}} + e_{2I} + \overline{e_{2I}}. To find a function p_{2} satisfying (3.7), it is enough to consider a weight n expression of the form:

(3.8)  p_{2}(z_{1}, z_{2}, z_{3}, z_{4}) = p_{2A} + p_{2B} + p_{2C} + p_{2D} + p_{2E},
p_{2A} = ∑_{α > 2γ, α even} p_{2A}^{αβγ}z_{1}^{α}z_{2}^{β}z_{3}^{γ}z_{4}
p_{2B} = ∑ p_{2B}^{βγ}z_{1}^{2γ}z_{2}^{β}z_{3}^{γ}z_{4}
p_{2C} = ∑_{α>2γ} p_{2C}^{αβγ}z_{1}^{α}z_{2}^{β}z_{3}^{γ}
p_{2D} = ∑ p_{2D}^{βγ}z_{1}^{2γ}z_{2}^{β}z_{3}^{γ}
p_{2E} = ∑_{α < 2γ, α odd} p_{2E}^{αβγ}z_{1}^{α}z_{2}^{β}z_{3}^{γ}.

Then Equation (3.7) becomes

0 = e_{2A} + \overline{e_{2A}} + e_{2B} + e_{2C} + \overline{e_{2C}}

+e_{2F} + \overline{e_{2F}} + e_{2G} + \overline{e_{2G}} + e_{2H} + \overline{e_{2H}} + e_{2I} + \overline{e_{2I}}

+\frac{1}{2i}( ∑_{α>2γ} p_{2A}^{αβγ}z_{1}^{α}x_{2}^{β}z̄_{1}^{2γ}(z_{1} + x_{2})z̄_{1} − ∑_{α>2γ} \overline{p_{2A}^{αβγ}}z̄_{1}^{α}x_{2}^{β}z_{1}^{2γ}(z̄_{1} + x_{2})z_{1})

+\frac{1}{2i}(∑ p_{2B}^{βγ}z_{1}^{2γ}x_{2}^{β}z̄_{1}^{2γ}(z_{1} + x_{2})z̄_{1} − ∑ \overline{p_{2B}^{βγ}}z̄_{1}^{2γ}x_{2}^{β}z_{1}^{2γ}(z̄_{1} + x_{2})z_{1})

+\frac{1}{2i}( ∑_{α>2γ} p_{2C}^{αβγ}z_{1}^{α}x_{2}^{β}z̄_{1}^{2γ} − ∑_{α>2γ} \overline{p_{2C}^{αβγ}}z̄_{1}^{α}x_{2}^{β}z_{1}^{2γ})

+\frac{1}{2i}(∑ p_{2D}^{βγ}z_{1}^{2γ}x_{2}^{β}z̄_{1}^{2γ} − ∑ \overline{p_{2D}^{βγ}}z̄_{1}^{2γ}x_{2}^{β}z_{1}^{2γ})

+\frac{1}{2i}( ∑_{α<2γ} p_{2E}^{αβγ}z_{1}^{α}x_{2}^{β}z̄_{1}^{2γ} − ∑_{α<2γ} \overline{p_{2E}^{αβγ}}z̄_{1}^{α}x_{2}^{β}z_{1}^{2γ}).

By construction, the existence of a (formal) solution p_{2} of this equation follows from a straightforward comparison of coefficients.

∑_{α>2γ} p_{2A}^{αβγ}z_{1}^{α}x_{2}^{β}z̄_{1}^{2γ}(z_{1} + x_{2})z̄_{1} = −2ie_{2F}
= −2i(1 + \frac{x_{2}}{z_{1}} )e_{2D}
∑ p_{2B}^{βγ}z_{1}^{2γ}x_{2}^{β}z̄_{1}^{2γ}(z_{1} + x_{2})z̄_{1} = −2ie_{2H}
= −i(1 + \frac{x_{2}}{z_{1}} )e_{2E}
∑_{α>2γ} p_{2C}^{αβγ}z_{1}^{α}x_{2}^{β}z̄_{1}^{2γ} = −2i(e_{2A} + \overline{e_{2I}})
= −2ie_{2A} + i\frac{x_{2}}{z̄_{1}} e_{2E}
∑ p_{2D}^{βγ}z_{1}^{2γ}x_{2}^{β}z̄_{1}^{2γ} − ∑ \overline{p_{2D}^{βγ}}z̄_{1}^{2γ}x_{2}^{β}z_{1}^{2γ} = −2ie_{2B}
∑_{α<2γ} p_{2E}^{αβγ}z_{1}^{α}x_{2}^{β}z̄_{1}^{2γ} = −2i(\overline{e_{2C}} + \overline{e_{2G}})
= −2i\overline{e_{2C}} + 2i\frac{x_{2}}{z̄_{1}} \overline{e_{2D}}.

This determines p_{2} by giving a formula for each coefficient, using a convention that e_{2}^{abc} = 0 if c < 0, and recalling the normalization (3.8), so that the terms that do not appear in (3.8) have coefficient 0:

p_{2A}^{αβγ} = −2ie_{2}^{α+1,2γ+1,β}
p_{2B}^{βγ} = −ie_{2}^{2γ+1,2γ+1,β}
p_{2C}^{αβγ} = −2ie_{2}^{α,2γ,β}, if α > 2γ + 1
= −2ie_{2}^{α,2γ,β} + ie_{2}^{2γ+1,2γ+1,β−1}, if α = 2γ + 1
p_{2D}^{βγ} = −ie_{2}^{2γ,2γ,β}
p_{2E}^{αβγ} = −2i\overline{e_{2}^{2γ,α,β}} + 2i\overline{e_{2}^{2γ+1,α,β−1}}.

As mentioned earlier, only the terms of weight less than 2n − 2 are of interest, and we will not consider the higher order terms or issues of convergence, and instead only try to find formal series solutions of formal series equations. The remaining unknowns, p_{1}, p_{3}, p_{4}, will depend on all three functions E_{2}, e_{3}, e_{4}. We can assume that the solution we want will be of the form:

(3.9)  p_{1} = p_{1}(z_{1}, z_{2}, z_{3}) = p_{1A} + p_{1B} + p_{1C} + p_{1D},
p_{1A} = ∑ p_{1A}^{γ}z_{3}^{γ},
p_{1B} = ∑_{α > 0, α even} p_{1B}^{αβγ}z_{1}^{α}z_{2}^{β}z_{3}^{γ},
p_{1C} = ∑_{β>0} p_{1C}^{βγ}z_{2}^{β}z_{3}^{γ},
p_{1D} = ∑_{α odd} p_{1D}^{αβγ}z_{1}^{α}z_{2}^{β}z_{3}^{γ},
p_{3} = p_{3A} + p_{3B},
p_{3A} = ∑ p_{3A}^{αβγ}z_{1}^{α}z_{2}^{β}z_{3}^{γ},
p_{3B} = ∑ p_{3B}^{αβγ}z_{1}^{α}z_{2}^{β}z_{3}^{γ}z_{4},
p_{4} = p_{4A} + p_{4B},
p_{4A} = ∑ p_{4A}^{αβγ}z_{1}^{α}z_{2}^{β}z_{3}^{γ},
p_{4B} = ∑ p_{4B}^{αβγ}z_{1}^{α}z_{2}^{β}z_{3}^{γ}z_{4},

where p_{1} has weight n − 1, and does not depend on z_{4}, and p_{3}, p_{4} have weight n. We next turn to the third equation from the system (3.6),

(3.10)  0 = e_{4}(z_{1}, z̄_{1}, x_{2}) + p_{4}(z_{1}, x_{2}, z̄_{1}^{2}, (z_{1} + x_{2})z̄_{1})
−(z_{1} + x_{2})\overline{p_{1}(z_{1}, x_{2}, z̄_{1}^{2})}
−z̄_{1}(p_{1}(z_{1}, x_{2}, z̄_{1}^{2}) + Re(p_{2}(z_{1}, x_{2}, z̄_{1}^{2}, (z_{1} + x_{2})z̄_{1}))).

Since e_{4} is given and p_{2} has already been found, we combine these to get a quantity f_{4}, and then break the series into subseries:

f_{4} = e_{4}(z_{1}, z̄_{1}, x_{2}) − \frac{z̄_{1}}{2} (p_{2}(z_{1}, x_{2}, z̄_{1}^{2}, (z_{1} + x_{2})z̄_{1}) + \overline{p_{2}(z_{1}, x_{2}, z̄_{1}^{2}, (z_{1} + x_{2})z̄_{1})})
= ∑ f_{4}^{abc}z_{1}^{a}z̄_{1}^{b}x_{2}^{c}
= f_{4A} + f_{4B} + f_{4C} + f_{4D} + f_{4E},
f_{4A} = ∑_{b even} f_{4}^{abc}z_{1}^{a}z̄_{1}^{b}x_{2}^{c},
f_{4B} = ∑_{a odd, b odd} f_{4}^{abc}z_{1}^{a}z̄_{1}^{b}x_{2}^{c},
f_{4C} = ∑_{a > 0, a even, b odd} f_{4}^{abc}z_{1}^{a}z̄_{1}^{b}x_{2}^{c},
f_{4D} = ∑_{c > 0, b odd} f_{4}^{0bc}z̄_{1}^{b}x_{2}^{c},
f_{4E} = ∑_{b odd} f_{4}^{0b0}z̄_{1}^{b}.

Note that e_{4} has degree n and the quantity \frac{z̄_{1}}{2} Re(p_{2}(z_{1}, x_{2}, z̄_{1}^{2}, (z_{1} + x_{2})z̄_{1})) has degree n + 1. Equation (3.10) then becomes

0 = f_{4A} + f_{4B} + f_{4C} + f_{4D} + f_{4E} + p_{4} − (z_{1} + x_{2})\overline{p_{1}} − z̄_{1}(p_{1A} + p_{1B} + p_{1C} + p_{1D}),

and by inspection, there are only two parts of this expression with z̄_{1}^{b} terms, so we can conclude

0 = f_{4E} − z̄_{1}p_{1A} = ∑_{b odd} f_{4}^{0b0}z̄_{1}^{b} − z̄_{1} ∑ p_{1A}^{γ}z̄_{1}^{2γ},

and this determines the coefficients of p_{1A}: p_{1A}^{γ} = f_{4}^{0,2γ+1,0}. The remaining linear equation from the system (3.6) is

(3.11)  0 = e_{3}(z_{1}, z̄_{1}, x_{2}) + p_{3}(z_{1}, x_{2}, z̄_{1}^{2}, (z_{1} + x_{2})z̄_{1}) − 2z̄_{1}\overline{p_{1}(z_{1}, x_{2}, z̄_{1}^{2})}

and we break e_{3} into subseries,

e_{3} = e_{3A} + e_{3B} + e_{3C} + e_{3D} + e_{3E},
e_{3A} = ∑_{b even} e_{3}^{abc}z_{1}^{a}z̄_{1}^{b}x_{2}^{c},
e_{3B} = ∑_{a odd, b odd} e_{3}^{abc}z_{1}^{a}z̄_{1}^{b}x_{2}^{c},
e_{3C} = ∑_{a even, b odd, b > 1} e_{3}^{abc}z_{1}^{a}z̄_{1}^{b}x_{2}^{c},
e_{3D} = ∑_{a even, c > 0} e_{3}^{a1c}z_{1}^{a}z̄_{1}x_{2}^{c},
e_{3E} = ∑_{a even, a > 0} e_{3}^{a10}z_{1}^{a}z̄_{1},

and we further rearrange the e_{3B} quantity:

e_{3B} = e_{3F} + e_{3G} + e_{3H},
e_{3F} = ∑_{a odd, b odd} e_{3}^{abc}z_{1}^{a−1}(z_{1} + x_{2})z̄_{1}^{b}x_{2}^{c},
e_{3G} = − ∑_{a odd, b odd, b > 1} e_{3}^{abc}z_{1}^{a−1}z̄_{1}^{b}x_{2}^{c+1},
e_{3H} = − ∑_{a odd} e_{3}^{a1c}z_{1}^{a−1}z̄_{1}x_{2}^{c+1}.

Equation (3.11) becomes

0 = e_{3A} + e_{3F} + e_{3G} + e_{3H} + e_{3C} + e_{3D} + e_{3E} + p_{3} − 2z̄_{1}(\overline{p_{1A}} + \overline{p_{1B}} + \overline{p_{1C}} + \overline{p_{1D}}), and the only terms with monomials of the form z_{1}^{a}z̄_{1}, with a even, are e_{3E} and the known quantity z̄_{1}\overline{p_{1A}}, so we collect these together and get three new expressions:

e_{3E} − 2z̄_{1}\overline{p_{1A}} = ∑_{a even, a > 0} f_{3}^{a}z_{1}^{a}z̄_{1}

= ((1 + \frac{x_{2}}{z_{1}} ) − (1 + \frac{x_{2}}{z_{1}} )\frac{x_{2}}{z_{1}} + (\frac{x_{2}}{z_{1}} )^{2})(e_{3E} − 2z̄_{1}\overline{p_{1A}})
= f_{3A} + f_{3B} + f_{3C},
f_{3A} = ∑_{a even, a > 0} f_{3}^{a}z_{1}^{a−1}(z_{1} + x_{2})z̄_{1}
f_{3B} = − ∑_{a even, a > 0} f_{3}^{a}z_{1}^{a−2}(z_{1} + x_{2})z̄_{1}x_{2}
f_{3C} = ∑_{a even, a > 0} f_{3}^{a}z_{1}^{a−2}z̄_{1}x_{2}^{2}.

Equation (3.11) becomes

0 = e_{3A} +e_{3F} +e_{3G} +e_{3H} +e_{3C} +e_{3D} +f_{3A} +f_{3B} +f_{3C} +p_{3} −2z̄_{1}(\overline{p_{1B}} +\overline{p_{1C}} +\overline{p_{1D}}).

The terms with monomials of the form z_{1}^{a}z̄_{1}^{b}x_{2}^{c} with a even, b odd, and b > 1 are e_{3G}, e_{3C}, and z̄_{1}\overline{p_{1B}}, and the terms with monomials of the form z_{1}^{a}z̄_{1}x_{2}^{c} with a even and c > 0 are e_{3H}, e_{3D}, f_{3C}, and z_{1}\overline{p_{1C}}, giving

0 = e_{3G} + e_{3C} − 2z̄_{1}\overline{p_{1B}}
= − ∑_{a odd, b odd, b > 1} e_{3}^{abc}z_{1}^{a−1}z̄_{1}^{b}x_{2}^{c+1} + ∑_{a even, b odd, b > 1} e_{3}^{abc}z_{1}^{a}z̄_{1}^{b}x_{2}^{c}

−2z̄_{1}\overline{∑_{α > 0, α even} p_{1B}^{αβγ}z_{1}^{α}x_{2}^{β}z̄_{1}^{2γ}},

0 = e_{3H} + e_{3D} + f_{3C} − 2z̄_{1}\overline{p_{1C}}
= − ∑_{a odd} e_{3}^{a1c}z_{1}^{a−1}z̄_{1}x_{2}^{c+1} + ∑_{a even, c > 0} e_{3}^{a1c}z_{1}^{a}z̄_{1}x_{2}^{c}

+ ∑_{a even, a > 0} f_{3}^{a}z_{1}^{a−2}z̄_{1}x_{2}^{2} − 2z̄_{1}\overline{∑_{β>0} p_{1C}^{βγ}x_{2}^{β}z̄_{1}^{2γ}}.

This determines the coefficients of p_{1B} and p_{1C}, again using the convention that e_{3}^{abc} = 0 if c < 0, and that the following formulas apply only to terms that appear in the normalization (3.9):

p_{1B}^{αβγ} = \frac{1}{2}\overline{e_{3}^{2γ,α+1,β}} − \frac{1}{2}\overline{e_{3}^{2γ+1,α+1,β−1}},
p_{1C}^{βγ} = \{ \begin{matrix}\frac{1}{2} \overline{e_{3}^{2γ,1,β}} − \frac{1}{2} \overline{e_{3}^{2γ+1,1,β−1}} & if β ≠ 2, \\ \frac{1}{2} \overline{e_{3}^{2γ,1,2}} − \frac{1}{2} \overline{e_{3}^{2γ+1,1,1}} + \frac{1}{2} \overline{f_{3}^{2γ+2}} & if β = 2. \end{matrix}

Returning to Equation (3.10),

0 = f_{4A} + f_{4B} + f_{4C} + f_{4D} + f_{4E} + p_{4} − (z_{1} + x_{2})\overline{p_{1}} − z̄_{1}(p_{1A} + p_{1B} + p_{1C} + p_{1D}),

the only terms with monomials of the form z_{1}^{a}z̄_{1}^{b}x_{2}^{c}, with a even and positive, and b odd, are f_{4C} and z̄_{1}p_{1B}, and the only terms with monomials of the form z̄_{1}^{b}x_{2}^{c}, with b odd and c positive, are f_{4D} and z̄_{1}p_{1C}. So, we collect these together and rearrange:

f_{4C} − z̄_{1}p_{1B} = ∑_{a even, a > 0, b odd} g_{4}^{abc}z_{1}^{a}z̄_{1}^{b}x_{2}^{c}
= (1 + \frac{x_{2}}{z_{1}} )(f_{4C} − z̄_{1}p_{1B}) − \frac{x_{2}}{z_{1}} (f_{4C} − z̄_{1}p_{1B}) = g_{4A} + g_{4B},
g_{4A} = ∑_{a even, a > 0, b odd} g_{4}^{abc}z_{1}^{a−1}(z_{1} + x_{2})z̄_{1}^{b}x_{2}^{c},
g_{4B} = − ∑_{a even, a > 0, b odd} g_{4}^{abc}z_{1}^{a−1}z̄_{1}^{b}x_{2}^{c+1},
f_{4D} − z̄_{1}p_{1C} = ∑_{c > 0, b odd} g_{4}^{0bc}z̄_{1}^{b}x_{2}^{c}
= (1 + \frac{z_{1}}{x_{2}} )(f_{4D} − z̄_{1}p_{1C}) − \frac{z_{1}}{x_{2}} (f_{4D} − z̄_{1}p_{1C}) = g_{4C} + g_{4D},
g_{4C} = ∑_{c > 0, b odd} g_{4}^{0bc}z̄_{1}^{b}(z_{1} + x_{2})x_{2}^{c−1},
g_{4D} = − ∑_{c > 0, b odd} g_{4}^{0bc}z_{1}z̄_{1}^{b}x_{2}^{c−1}.

Note that the last rearrangement is different from the previous ones. Equation (3.10) then becomes

0 = f_{4A} + f_{4B} + g_{4A} + g_{4B} + g_{4C} + g_{4D} + f_{4E} + p_{4A} + p_{4B}
−(z_{1} + x_{2})(\overline{p_{1A}} + \overline{p_{1B}} + \overline{p_{1C}} + \overline{p_{1D}}) − z̄_{1}(p_{1A} + p_{1D}).

The only terms with monomials of the form z_{1}^{a}z̄_{1}^{b}x_{2}^{c}, with a and b both odd, are f_{4B}, g_{4B}, g_{4D}, and z̄_{1}p_{1D}, and collecting these terms gives

0 = f_{4B} + g_{4B} + g_{4D} − z̄_{1}p_{1D}
= ∑_{a odd, b odd} f_{4}^{abc}z_{1}^{a}z̄_{1}^{b}x_{2}^{c} − ∑_{a even, a > 0, b odd} g_{4}^{abc}z_{1}^{a−1}z̄_{1}^{b}x_{2}^{c+1}

− ∑_{c > 0, b odd} g_{4}^{0bc}z_{1}z̄_{1}^{b}x_{2}^{c−1} − ∑_{α odd} p_{1D}^{αβγ}z_{1}^{α}x_{2}^{β}z̄_{1}^{2γ+1},

which determines p_{1D}:

p_{1D}^{αβγ} = \{ \begin{matrix}f_{4}^{α,2γ+1,β} − g_{4}^{α+1,2γ+1,β−1} & if α > 1, \\ f_{4}^{1,2γ+1,β} − g_{4}^{2,2γ+1,β−1} − g_{4}^{0,2γ+1,β+1} & if α = 1. \end{matrix}

The only remaining unknowns in this equation are p_{4A} and p_{4B}. The terms involving monomials of the form z_{1}^{a}x_{2}^{b}(z_{1} + x_{2})z̄_{1}^{b}, with b odd, are g_{4A}, g_{4C}, the known quantity (z_{1} + x)\overline{p_{1D}}, and p_{4B}, and collecting these terms gives

0 = g_{4A} + g_{4C} − (z_{1} + x_{2})\overline{p_{1D}} + p_{4B}
= ∑_{a even, a > 0, b odd} g_{4}^{abc}z_{1}^{a−1}(z_{1} + x_{2})z̄_{1}^{b}x_{2}^{c} + ∑_{c > 0, b odd} g_{4}^{0bc}z̄_{1}^{b}(z_{1} + x_{2})x_{2}^{c−1}

−(z_{1} + x_{2}) ∑_{α odd} \overline{p_{1D}^{αβγ}z_{1}^{α}x_{2}^{β}z̄_{1}^{2γ}} + ∑ p_{4B}^{αβγ}z_{1}^{α}x_{2}^{β}z̄_{1}^{2γ}(z_{1} + x_{2})z̄_{1},

which determines p_{4B}. The only terms left are those with even powers of z̄_{1}:

0 = f_{4A} + p_{4A} − (z_{1} + x_{2})(\overline{p_{1A}} + \overline{p_{1B}} + \overline{p_{1C}})
= ∑_{b even} f_{4}^{abc}z_{1}^{a}z̄_{1}^{b}x_{2}^{c} + ∑ p_{4A}^{αβγ}z_{1}^{α}x_{2}^{β}z̄_{1}^{2γ}

−(z_{1} + x_{2}) (\overline{∑ p_{1A}^{γ}z̄_{1}^{2γ}} + \overline{∑_{α > 0, α even} p_{1B}^{αβγ}z_{1}^{α}x_{2}^{β}z̄_{1}^{2γ}} + \overline{∑_{β>0} p_{1C}^{βγ}x_{2}^{β}z̄_{1}^{2γ}} ) ,

which determines p_{4A}.

Similarly for (3.11), the only remaining unknowns are p_{3A} and p_{3B}. The terms involving monomials of the form z_{1}^{a}x_{2}^{b}(z_{1} + x_{2})z̄_{1}^{b}, with b odd, are e_{3F}, f_{3A}, f_{3B}, and p_{3B}, and collecting these terms gives

0 = e_{3F} + f_{3A} + f_{3B} + p_{3B}
= ∑_{a odd, b odd} e_{3}^{abc}z_{1}^{a−1}(z_{1} + x_{2})z̄_{1}^{b}x_{2}^{c} + ∑_{a even, a > 0} f_{3}^{a}z_{1}^{a−1}(z_{1} + x_{2})z̄_{1}

− ∑_{a even, a > 0} f_{3}^{a}z_{1}^{a−2}(z_{1} + x_{2})z̄_{1}x_{2} + ∑ p_{3B}^{αβγ}z_{1}^{α}x_{2}^{β}z̄_{1}^{2γ}(z_{1} + x_{2})z̄_{1},

which determines p_{3B}. The only terms left are those with even powers of z̄_{1}:

0 = e_{3A} + p_{3A} − 2z̄_{1}\overline{p_{1D}}
= ∑_{b even} e_{3}^{abc}z_{1}^{a}z̄_{1}^{b}x_{2}^{c} + ∑ p_{3A}^{αβγ}z_{1}^{α}x_{2}^{β}z̄_{1}^{2γ} − 2z̄_{1}\overline{∑_{α odd} p_{1D}^{αβγ}z_{1}^{α}x_{2}^{β}z̄_{1}^{2γ}},

which determines p_{3A}.

The conclusion is that there exist formal series p_{1}, p_{2}, p_{3}, p_{4} that solve the system (3.6), for any given E_{2}, e_{3}, e_{4}, and that the coefficients of the solutions are complex linear combinations of the coefficients of E_{2}, e_{3}, e_{4} and their complex conjugates. As remarked at the beginning of the Proof, since the linear system has a formal solution, the non-linear system (3.5) has a formal solution also.

Conjecture 3.2. Given M with the non-degenerate quadratic normal form (type (I), as in (3.1)), there exist convergent series p_{1}, p_{2}, p_{3}, p_{4} which are solutions of the system of equations (3.5), so that M is analytically equivalent to the polynomial model.

Remark. As evidence in favor of the conjecture, note that among all the algebraic manipulations of the e⃗ series in the above Proof, including the decompositions into subseries, none drastically shrinks the radius of convergence. The form of the solution p⃗ of the linear problem suggests that its domain contains some polydisc in ℂ^{4} with radius lengths that depend on the radius lengths of the polydisc of convergence of the given quantity e⃗, in such a way that the new defining equations in the transformed coordinates will have a domain not much smaller than the domain of the original equations (after applying a suitable version of Lemma 2.3). When similar situations have arisen in other normal form problems ([M], [C_{2}]), the technique of rapid convergence has been used to prove that the sequence of compositions of approximate solutions converges to a holomorphic normalizing transformation on an open neighborhood of 0.

4. Some degenerate cases

In this Section we consider M satisfying the first non-degeneracy condition but not the second, so after a coordinate transformation putting the quadratic terms into normal form (II), the defining equations become

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

where E_{2}, e_{3}, e_{4} have degree 3. This differs from the non-degenerate case only in that it is missing a z̄_{1}x_{2} term.

Theorem 4.1. Given a manifold M with the type (II) normal form, exactly one of the following two cases must hold:

(1) There exists a unique integer N ≥ 3 and a holomorphic coordinate change such that the defining equations in the new coordinates are

y_{2} = O(N + 1)
z_{3} = z̄_{1}^{2} + O(N + 1)
z_{4} = z_{1}z̄_{1} + x_{2}^{N−1}z̄_{1} + O(N + 1).

(2) For any integer n ≥ 2, there exists a holomorphic coordinate change such that the defining equations in the new coordinates are

y_{2} = O(n + 1)
z_{3} = z̄_{1}^{2} + O(n + 1)
z_{4} = z_{1}z̄_{1} + O(n + 1).

In case (2), M is formally equivalent to the real algebraic variety {y_{2} = 0, z_{3} = z̄_{1}^{2}, z_{4} = z_{1}z̄_{1}}. It could be denoted the “N = ∞” case.

Proof. By hypothesis, there is some N ≥ 3 and some coordinate system where the defining equations for M are:

y_{2} = E_{2},
z_{3} = z̄_{1}^{2} + e_{3}
z_{4} = z_{1}z̄_{1} + e_{4},

where (E_{2}, e_{3}, e_{4}) has degree N. A coordinate transformation of the form (2.3), where p_{1} has weight N − 1, and p_{2}, p_{3}, p_{4} have weight N, gives:

Im(z̃_{2}) = Im(z_{2} + p_{2}) = E_{2} + Im(p_{2}(z_{1}, x_{2} + iE_{2}, z̄_{1}^{2} + e_{3}, z_{1}z̄_{1} + e_{4}))
= E_{2} + Im(p_{2}(z_{1}, x_{2}, z̄_{1}^{2}, z_{1}z̄_{1})) + O(2N − 1)
z̃_{3} − ¯ z̃_{1}^{2} = e_{3} + p_{3} − 2z̄_{1}\overline{p_{1}} − \overline{p_{1}}^{2}
= e_{3} + p_{3}(z_{1}, x_{2}, z̄_{1}^{2}, z_{1}z̄_{1}) − 2z̄_{1}\overline{p_{1}(z_{1}, x_{2}, z̄_{1}^{2}, z_{1}z̄_{1})} + O(2N − 2)
z̃_{4} − z̃_{1} ¯z̃_{1} = e_{4} + p_{4} − z_{1}\overline{p_{1}} − z̄_{1}p_{1} − p_{1}\overline{p_{1}}
= e_{4} + p_{4}(z_{1}, x_{2}, z̄_{1}^{2}, z_{1}z̄_{1}) − z̄_{1}p_{1}(z_{1}, x_{2}, z̄_{1}^{2}, z_{1}z̄_{1})
−z_{1}\overline{p_{1}(z_{1}, x_{2}, z̄_{1}^{2}, z_{1}z̄_{1})} + O(2N − 2).

Any term in E_{2} of the form z_{1}^{a}z̄_{1}^{b}x_{2}^{c} with a > b can be eliminated by some term of p_{2} of the form z_{1}^{a}z_{2}^{c}z_{3}^{b/2} if b is even, or z_{1}^{a−1}z_{2}^{c}z_{3}^{(b−1)/2}z_{4} if b is odd. Since E_{2} is real-valued, the a > b terms are matched by their complex conjugates, which are canceled by \overline{p_{2}}. Similarly, the z_{1}^{a}z̄_{1}^{a}x_{2}^{c} terms, with real coefficients, can also be eliminated by p_{2} terms with a either even or odd.

Consider p_{1} of the form p_{1A} + p_{1B} with p_{1A} = ∑ p_{1A}^{jk}z_{1}^{2j}z_{2}^{k}, p_{1B} = ∑ p_{1B}^{jk}z_{2}^{k}z_{3}^{j>1}. The degree N terms of e_{3} that have the form z̄_{1}^{odd}x_{2}^{k} can be eliminated by p_{1A} terms of the form z_{1}^{odd−1}z_{2}^{k}. Then the degree N terms of e_{4} − z̄_{1}p_{1A} that have the form z̄_{1}^{odd}x_{2}^{k} can be eliminated by p_{1B} terms of the form z_{2}^{k}z_{4}^{(odd−1)/2}, except for the z̄_{1}x_{2}^{N−1} term, since that component of p_{1} was already used, and the remaining terms of e_{4} − z_{1}\overline{p_{1}} − z̄_{1}p_{1A} can be eliminated by p_{4}. The remaining terms of e_{3} − 2z̄_{1}\overline{p_{1B}}, none of which are of the form z̄_{1}^{odd}x_{2}^{k}, can be eliminated by p_{3}. A linear transformation of the form z̃_{1} = c_{1}z_{1}, z̃_{2} = z_{2}, z̃_{3} = \overline{c_{1}}^{2}z_{3}, z̃_{4} = c_{1}\overline{c_{1}}z_{4}, will not alter the coefficients on the quadratic terms, but can normalize the coefficient of z̄_{1}x_{2}^{N−1} in e_{4} to either 1, in which case the existence of N in Case (1) is established, or else 0, in which case we return to the beginning of the Proof with N increased by one, to establish either Case (1) for some larger N, or Case (2) by induction.

For the uniqueness from Case (1), suppose the defining equations are of the form

y_{2} = E_{2},
z_{3} = z̄_{1}^{2} + e_{3}
z_{4} = z_{1}z̄_{1} + x_{2}^{N−1}z̄_{1} + e_{4},

where (E_{2}, e_{3}, e_{4}) has degree N + 1. A coordinate transformation of the form (2.3), with p_{1} weight 2, p_{2}, p_{3}, p_{4} weight 3, gives:

Im(z̃_{2}) = Im(z_{2} + p_{2})
= E_{2} + Im(p_{2}(z_{1}, x_{2} + iE_{2}, z̄_{1}^{2} + e_{3}, z_{1}z̄_{1} + x_{2}^{N−1}z̄_{1} + e_{4}))
= E_{2} + Im(p_{2}(z_{1}, x_{2}, z̄_{1}^{2}, z_{1}z̄_{1})) + O(N + 1)
z̃_{3} − ¯ z̃_{1}^{2} = e_{3} + p_{3} − 2z̄_{1}\overline{p_{1}} − \overline{p_{1}}^{2}
= e_{3} + p_{3}(z_{1}, x_{2}, z̄_{1}^{2}, z_{1}z̄_{1}) − 2z̄_{1}\overline{p_{1}(z_{1}, x_{2}, z̄_{1}^{2}, z_{1}z̄_{1})} − \overline{p_{1}}^{2} + O(N + 1)
z̃_{4} − z̃_{1} ¯z̃_{1} = x_{2}^{N−1}z̄_{1} + e_{4} + p_{4} − z_{1}\overline{p_{1}} − z̄_{1}p_{1} − p_{1}\overline{p_{1}}
= x_{2}^{N−1}z̄_{1} + e_{4} + p_{4}(z_{1}, x_{2}, z̄_{1}^{2}, z_{1}z̄_{1}) − z̄_{1}p_{1}(z_{1}, x_{2}, z̄_{1}^{2}, z_{1}z̄_{1})

−z_{1}\overline{p_{1}(z_{1}, x_{2}, z̄_{1}^{2}, z_{1}z̄_{1})} − p_{1}\overline{p_{1}} + O(N + 1). (4.2)

If there were some coordinate change taking the degree N normal form to the normal form of some other degree N′, we can assume (by switching) that N′ > N, so the coordinate change would eliminate all the terms of degree ≤ N, including the z̄_{1}x_{2}^{N−1} term, from (4.2). Note that to get to (4.2) from the line before it, we are replacing p_{k}(z_{1}, x_{2} + iE_{2}, z̄_{1}^{2} + e_{3}, z_{1}z̄_{1} + x_{2}^{N−1}z̄_{1} + e_{4}) by p_{k}(z_{1}, x_{2}, z̄_{1}^{2}, z_{1}z̄_{1}), but even when p_{k} has weight less than N, the difference between these quantities has degree at least N + 1. The previous part of the Proof showed that the z_{2}^{k} term of p_{1} is the only term out of p_{1}, p_{2}, p_{3}, p_{4} that can eliminate a z̄_{1}x_{2}^{N−1} term from e_{3} or e_{4} from the simplified equations, and it cannot always do both. However, to establish uniqueness, we must consider the non-linear components also — there is a possibility that some lower weight terms of p⃗ could contribute to the z̄_{1}x_{2}^{N−1} term. Considering the products \overline{p_{1}}^{2}, and p_{1}\overline{p_{1}}, the only way a term of the form z̄_{1}x_{2}^{N−1} could appear is if there were terms p_{1}^{a}z_{2}^{a} and p_{1}^{b}z_{1}z_{2}^{b}, with a + b = N − 1. However, a non-zero coefficient on the p_{1}^{a}z_{2}^{a} term would contribute a term of the form z̄_{1}x_{2}^{a} for some a < N − 1 to both the e_{3} and e_{4} expressions, which could not be canceled by any of the other terms from any p_{k}.

It follows from the above Proof that for manifolds with a type (II) quadratic normal form, the value of N, 3 ≤ N ≤ ∞, is an invariant of M under formal coordinate changes, and each equivalence class indexed by 3 ≤ N ≤ ∞ is obviously non-empty, with a polynomial model as a representative manifold.

Theorem 4.2. Given N ≥ 3, suppose a manifold M is in the normal form from Case (1) of Theorem 4.1:

y_{2} = O(N + 1)
z_{3} = z̄_{1}^{2} + O(N + 1)
z_{4} = z_{1}z̄_{1} + x_{2}^{N−1}z̄_{1} + O(N + 1).

Then, for any n ≥ N, there exists a holomorphic coordinate change such that the defining equations in the new coordinates are

y_{2} = O(n + 1)
z_{3} = z̄_{1}^{2} + O(n + 1)
z_{4} = z_{1}z̄_{1} + x_{2}^{N−1}z̄_{1} + O(n + 1).

That is, M is formally equivalent to the real algebraic variety {y_{2} = 0, z_{3} = z̄_{1}^{2}, z_{4} = z_{1}z̄_{1} + x_{2}^{N−1}z̄_{1}}.

Proof. The n = N case is trivial and also the start of an induction: we will show that if the defining equations are of the form

y_{2} = E_{2}(z_{1}, z̄_{1}, x_{2})
z_{3} = z̄_{1}^{2} + e_{3}(z_{1}, z̄_{1}, x_{2})
z_{4} = z_{1}z̄_{1} + x_{2}^{N−1}z̄_{1} + e_{4}(z_{1}, z̄_{1}, x_{2}).

with (E_{2}, e_{3}, e_{4}) degree n > N, then there exists a transformation of the form (2.3) which eliminates the degree n terms, so that the (E_{2}, e_{3}, e_{4}) expression in the new coordinates has degree at least n + 1.

For points z⃗ = (z_{1}, x_{2} + iE_{2}, z̄_{1}^{2} + e_{3}, (z_{1} + x_{2}^{N−1})z̄_{1} + e_{4}) on M,

Im(z̃_{2}) = Im(z_{2} + p_{2}(z⃗))
= E_{2}(z_{1}, z̄_{1}, x_{2}) + Im(p_{2}(z⃗)),
z̃_{3} − ¯ z̃_{1}^{2} = z_{3} + p_{3}(z⃗) − \overline{(z_{1} + p_{1}(z⃗))}^{2}
(4.3)  = e_{3} + p_{3} − 2z̄_{1}\overline{p_{1}} − \overline{p_{1}}^{2},
z̃_{4} − (z̃_{1} + x̃_{2}^{N−1})¯z̃_{1}
= z_{4} + p_{4}(z⃗) − (z_{1} + p_{1}(z⃗) + (x_{2} + Re(p_{2}(z⃗)))^{N−1})\overline{(z_{1} + p_{1}(z⃗))}
= e_{4} + p_{4} − (z_{1} + x_{2}^{N−1})\overline{p_{1}} − z̄_{1}p_{1} − z̄_{1}((x_{2} + Re(p_{2}))^{N−1} − x_{2}^{N−1})

−p_{1}\overline{p_{1}} − ((x_{2} + Re(p_{2}))^{N−1} − x_{2}^{N−1})\overline{p_{1}}. (4.4)

Since we only want to eliminate the degree n terms, we will be able to ignore any expressions of degree ≥ n + 1. However, the inhomogeneity of the “normal form” part of the equations makes this more complicated than the situation from Section 3. In particular, we will need some of the terms of p⃗ to have weight less than n.

To start, we make the following choices for the form of the coordinate transformation:

p_{1}(z_{1}, z_{2}, z_{3}) = p_{1A}z_{2}^{n−1} + p_{1B}(z_{1}, z_{2}) + p_{1C}z_{1}z_{2}^{n−N} + p_{1D}(z_{2}, z_{3})
p_{1B} = ∑_{k+2j=n−1,j>0} p_{1B}^{jk}z_{1}^{2j}z_{2}^{k}
p_{1D} = ∑_{k+2j=n−1,j>0} p_{1D}^{kj}z_{2}^{k}z_{3}^{j}
p_{2}(z_{1}, z_{2}, z_{3}, z_{4}) = p_{2A}(z_{1}, z_{2}, z_{3}) + z_{4}p_{2B}(z_{1}, z_{2}, z_{3})
p_{3}(z_{1}, z_{2}, z_{3}, z_{4}) = p_{3A}z_{2}^{n−N}z_{3} + p_{3B}z_{2}^{2(n−N)}z_{3}
+p_{3C}(z_{1}, z_{2}, z_{3}) + z_{4}p_{3D}(z_{1}, z_{2}, z_{3})
p_{4}(z_{1}, z_{2}, z_{3}, z_{4}) = p_{4A}z_{2}^{n−N}z_{4} + p_{4B}z_{2}^{2(n−N)}z_{4}
+p_{4C}(z_{1}, z_{2}, z_{3}) + z_{4}p_{4D}(z_{1}, z_{2}, z_{3}).

The weight of p_{1C} is n − N + 1, which is in the interval [2, n − 2], and the weights of p_{3A} and p_{4A} are one higher, n − N + 2. The weights of p_{3B} and p_{4B} are each 2(n − N + 1), which is in the interval [4, 2n − 4]. The p_{1A}, p_{1B}, p_{1D} parts of p_{1} have weight n − 1, and the p_{2}, p_{3C} + z_{4}p_{3D} and p_{4C} + z_{4}p_{4D} quantities are chosen to have weight n. If the weight, 2(n − N + 1), of p_{3B} and p_{4B} is greater than n, these quantities can be ignored for the purposes of this Proof. If 2(n − N + 1) = n, then a z_{2}^{2(n−N)}z_{3} term could be included in the p_{3C} expression and a z_{2}^{2(n−N)}z_{4} in the z_{4}p_{3D} expression, so there is again no need to consider separately the p_{3B} and p_{4B} terms.

Considering the lowest weight quantity, the difference between p_{1C}z_{1}(x_{2}+iE_{2})^{n−N} and p_{1C}z_{1}x_{2}^{n−N} has degree (n − N + 1) − 1 + n = 2n − N ≥ n + 1. Similarly, the differences p_{3A}(x_{2} + iE_{2})^{n−N}(z̄_{1}^{2} + e_{3}) − p_{3A}x_{2}^{n−N}z̄_{1}^{2} and p_{4A}(x_{2} + iE_{2})^{n−N}(z_{1}z̄_{1} + x_{2}^{N−1}z̄_{1} +e_{4})−p_{4A}x_{2}^{n−N}(z_{1}z̄_{1} +x_{2}^{N−1}z̄_{1}) have degree (n−N +2)−2+n = 2n−N ≥ n + 1. The E_{2}, e_{3}, e_{4} quantities similarly contribute only high degree terms to the expressions with higher weights, so in (4.3) and (4.4), we can replace z⃗ with (z_{1}, x_{2}, z̄_{1}^{2}, (z_{1} + x_{2}^{N−1})z̄_{1}), and include the differences in a “+O(n + 1)” quantity.

The quantity ((x_{2}+Re(p_{2}(z⃗)))^{N−1}−x^{N−1}) from (4.4) has degree N−2+n ≥ n+1, so the terms with that as a factor can also be included in the O(n + 1) quantity. Expanding \overline{p_{1}(z_{1}, x_{2} + iE_{2}, z_{1}z̄_{1} + e_{3})}^{2}, the only term that might have degree n or

less is \overline{(p_{1C}z_{1}x_{2}^{n−N})}^{2} and similarly, the only term of

p_{1}(z_{1}, x_{2} + iE_{2}, z_{1}z̄_{1} + e_{3})\overline{p_{1}(z_{1}, x_{2} + iE_{2}, z_{1}z̄_{1} + e_{3})}

that might have degree n or less is (p_{1C}z_{1}x_{2}^{n−N})\overline{(p_{1C}z_{1}x_{2}^{n−N})}. All the other parts of the products have higher degree, such as \overline{p_{1A}p_{1C}}, which has degree (n−1)+(n− N + 1) = 2n − N ≥ n + 1. There is also a term x_{2}^{N−1}\overline{p_{1}(z⃗)} in (4.4), and the part that has the form

x_{2}^{N−1}\overline{(p_{1A}(x_{2} + iE_{2})^{n−1} + p_{1B}(z_{1}, x_{2} + iE_{2}) + p_{1D}(x_{2} + iE_{2}, z_{1}z̄_{1} + e_{3}))}

has degree N − 1 + n − 1 ≥ n + 1. So, (4.3) and (4.4) simplify to:

z̃_{3} − ¯ z̃_{1}^{2} = e_{3}(z_{1}, z̄_{1}, x_{2}) + p_{3}(z_{1}, x_{2}, z̄_{1}^{2}, (z_{1} + x_{2}^{N−1})z̄_{1})
−2z̄_{1}\overline{p_{1}(z_{1}, x_{2}, z̄_{1}^{2})} − \overline{(p_{1C}z_{1}x_{2}^{n−N})}^{2} + O(n + 1),

z̃_{4} − (z̃_{1} + x̃_{2}^{N−1})¯z̃_{1} = e_{4}(z_{1}, z̄_{1}, x_{2}) + p_{4}(z_{1}, x_{2}, z̄_{1}^{2}, (z_{1} + x_{2}^{N−1})z̄_{1})
−z_{1}\overline{p_{1}(z_{1}, x_{2}, z̄_{1}^{2})} − z̄_{1}p_{1}(z_{1}, x_{2}, z̄_{1}^{2})
−x_{2}^{N−1}\overline{(p_{1C}z_{1}x_{2}^{n−N})}
−(p_{1C}z_{1}x_{2}^{n−N})\overline{(p_{1C}z_{1}x_{2}^{n−N})} + O(n + 1).

When substituted into the weight n expressions z_{4}p_{2B}(z_{1}, z_{2}, z_{3}), z_{4}p_{3D}(z_{1}, z_{2}, z_{3}), and z_{4}p_{4D}(z_{1}, z_{2}, z_{3}), the quantity z_{4} = z_{1}z̄_{1} + x_{2}^{N−1}z̄_{1} + e_{4} can be replaced with z_{1}z̄_{1}, since the difference has degree n − 2 + N ≥ n + 1. Similarly, p_{4B}(x_{2} + iE_{2})^{2(n−N)}(z_{1}z̄_{1}+x_{2}^{N−1}z̄_{1}+e_{4}) differs from p_{4B}x_{2}^{2(n−N)}z_{1}z̄_{1} only by a quantity with degree n+1. However, in the weight n−N +2 expression p_{4A}(x_{2} +iE_{2})^{n−N}(z_{1}z̄_{1} + x_{2}^{N−1}z̄_{1} +e_{4}), the x_{2}^{N−1}z̄_{1} part does contribute a degree n term, so we can’t replace z_{4} with just z_{1}z̄_{1}, but the difference p_{4A}(x_{2} + iE_{2})^{n−N}(z_{1}z̄_{1} + x_{2}^{N−1}z̄_{1} + e_{4}) − p_{4A}x_{2}^{n−N}(z_{1}z̄_{1} + x_{2}^{N−1}z̄_{1}) has degree 2n − N ≥ n + 1.

Finally, breaking e_{3} and e_{4} into convenient subseries, we get the simplified (but still non-linear if 2(n − N) + 2 ≤ n) expressions:

Im(z̃_{2}) = E_{2}(z_{1}, z̄_{1}, x_{2}) + Im(p_{2}(z_{1}, x_{2}, z̄_{1}^{2}, z_{1}z̄_{1})) + O(n + 1),
z̃_{3} − ¯ z̃_{1}^{2}
(4.5) = ∑_{b even} e_{3}^{abc}z_{1}^{a}z̄_{1}^{b}x_{2}^{c} + p_{3C}(z_{1}, x_{2}, z̄_{1}^{2})

+ ∑_{b odd, a > 0} e_{3}^{abc}z_{1}^{a}z̄_{1}^{b}x_{2}^{c} − 2z̄_{1}\overline{∑_{j>0} p_{1D}^{kj}x_{2}^{k}z̄_{1}^{2j}} + z_{1}z̄_{1}p_{3D}(z_{1}, x_{2}, z̄_{1}^{2}) (4.6)

+ ∑_{b odd, b > 1} e_{3}^{0bc}z̄_{1}^{b}x_{2}^{c} − 2z̄_{1}\overline{∑_{j>0} p_{1B}^{jk}z_{1}^{2j}x_{2}^{k}} (4.7)

+e_{3}^{0,1,n−1}z̄_{1}x_{2}^{n−1} − 2z̄_{1}\overline{(p_{1A}x_{2}^{n−1})} (4.8)

+p_{3A}x_{2}^{n−N}z̄_{1}^{2} − 2z̄_{1}\overline{(p_{1C}z_{1}x_{2}^{n−N})} (4.9)

+p_{3B}x_{2}^{2(n−N)}z̄_{1}^{2} − \overline{(p_{1C}z_{1}x_{2}^{n−N})}^{2} + O(n + 1), (4.10)

z̃_{4} − (z̃_{1} + x̃_{2}^{N−1})¯z̃_{1}
= ∑_{b even} e_{4}^{abc}z_{1}^{a}z̄_{1}^{b}x_{2}^{c} − z_{1}\overline{(p_{1A}x_{2}^{n−1})} − z_{1}\overline{∑_{j>0} p_{1B}^{jk}z_{1}^{2j}x_{2}^{k}}

−z_{1}\overline{∑_{j>0} p_{1D}^{kj}x_{2}^{k}z̄_{1}^{2j}} + p_{4C}(z_{1}, x_{2}, z̄_{1}^{2}) (4.11)

+ ∑_{b odd, a > 0} e_{4}^{abc}z_{1}^{a}z̄_{1}^{b}x_{2}^{c} − z̄_{1} ∑_{j>0} p_{1B}^{jk}z_{1}^{2j}x_{2}^{k} + z_{1}z̄_{1}p_{4D}(z_{1}, x_{2}, z̄_{1}^{2}) (4.12)

+ ∑_{b odd, b > 1} e_{4}^{0bc}z̄_{1}^{b}x_{2}^{c} − z̄_{1} ∑_{j>0} p_{1D}^{kj}x_{2}^{k}z̄_{1}^{2j} (4.13)

+e_{4}^{0,1,n−1}z̄_{1}x_{2}^{n−1} − z̄_{1}p_{1A}x_{2}^{n−1} − x_{2}^{N−1}\overline{(p_{1C}z_{1}x_{2}^{n−N})} + p_{4A}x_{2}^{n−N}x_{2}^{N−1}z̄_{1} (4.14)

+p_{4A}x_{2}^{n−N}z_{1}z̄_{1} − z̄_{1}p_{1C}z_{1}x_{2}^{n−N} − z_{1}\overline{(p_{1C}z_{1}x_{2}^{n−N})} (4.15)

+p_{4B}x_{2}^{2(n−N)}z_{1}z̄_{1} − (p_{1C}z_{1}x_{2}^{n−N})\overline{(p_{1C}z_{1}x_{2}^{n−N})} + O(n + 1), (4.16)

We note first that it is easy to find p_{2}(z_{1}, z_{2}, z_{3}, z_{4}) such that in a coordinate system where z̃_{2} = z_{2} + p_{2}, Im(z̃_{2}) = O(n + 1). As in the Proof of Theorem 4.1, any (formally) real-valued series E_{2}(z_{1}, z̄_{1}, x_{2}) can be canceled by an expression of the form Im(p_{2}(z_{1}, x_{2}, z̄_{1}^{2}, z_{1}z̄_{1})).

The other two expressions above have been grouped together to see how to choose the coefficients of p⃗. So that there are no terms of degree < n, we have to choose p_{3A} = 2\overline{p_{1C}} to eliminate line (4.9) and p_{4A} = p_{1C} + \overline{p_{1C}} for line (4.15). If 2n − 2N + 2 < n, we have to choose p_{3B} = \overline{p_{1C}}^{2} to eliminate line (4.10) and p_{4B} = p_{1C} \overline{p_{1C}} for line (4.16). If 2n − 2N + 2 > n, the quadratic quantities can be put with the O(n + 1) (as previously mentioned, there is no need for p_{3B}, p_{4B}). If

2n − 2N + 2 = n, then \overline{(p_{1C}z_{1}x_{2}^{n−N})}^{2} has the same form as the terms in (4.5) and (p_{1C}z_{1}x_{2}^{n−N})\overline{(p_{1C}z_{1}x_{2}^{n−N})} has the same form as the terms in (4.12).

To cancel the degree n terms in (4.8), p_{1A} = \frac{1}{2} \overline{e_{3}^{0,1,n−1}}, and p_{1B} is similarly determined by (4.7). The choices for p_{1A} and p_{4A} turn (4.14) into

(e_{4}^{0,1,n−1} − \frac{1}{2}\overline{e_{3}^{0,1,n−1}} − \overline{p_{1C}} + (p_{1C} + \overline{p_{1C}}))z̄_{1}x_{2}^{n−1},

so p_{1C} = \frac{1}{2} \overline{e_{3}^{0,1,n−1}} − e_{4}^{0,1,n−1}. Line (4.13) determines p_{1D}, and then the remaining lines, (4.5), (4.6), (4.11), (4.12) (including the quadratic terms if 2n − 2N + 2 = n) can clearly be eliminated by some p_{3C}, p_{3D}, p_{4C}, p_{4D}.

Remark. In the N ≥ 3 cases, no conjecture is offered on the analytic equivalence of M and the real algebraic model. Unlike the approximate solution from N = 2 case of the previous Section, which nearly doubled the degree of the defining equations e⃗, the above construction only increases the degree of the defining equation by one. The construction does not seem to be as well-suited as that of the N = 2 case for a straightforward application of the rapid convergence technique.

5. Conclusion

For N ≥ 2, let M_{N} denote the following real algebraic affine subvariety of ℂ^{n}:

M_{N} = {y_{2} = 0, z_{3} = z̄_{1}^{2}, z_{4} = z_{1}z̄_{1} + z̄_{1}x_{2}^{N−1}}.

Note that M_{N} is totally real at every point except the origin, where there is a CR singularity with type (I) if N = 2, or type (II) if N > 2. Similarly, denote

M_{∞} = {y_{2} = 0, z_{3} = z̄_{1}^{2}, z_{4} = z_{1}z̄_{1}}.

This variety has the structure of a product S × ℝ ⊆ ℂ^{3} × ℂ, where S is the CR singular real surface {(z_{1}, z_{3}, z_{4}) : z_{3} = z̄_{1}^{2}, z_{4} = z_{1}z̄_{1}} ⊆ ℂ^{3}, considered in [C_{2}]. M_{∞} has a CR singularity of type (II) at every point on the real line {(0, x_{2}, 0, 0)}, and is totally real at points not on the line.

The results of Sections 2, 3, 4 can be summarized as follows.

Proposition 5.1. Given a real analytic threefold M in ℂ^{4} with a CR singularity at 0, exactly one of the following cases holds:

(1) There exists a unique integer N ≥ 2 such that for any n ≥ N, there exists a biholomorphic coordinate transformation of a neighborhood of 0, so that in the new coordinate system, the defining equations of M are

y_{2} = O(n + 1)
z_{3} = z̄_{1}^{2} + O(n + 1)
z_{4} = z_{1}z̄_{1} + z̄_{1}x_{2}^{N−1} + O(n + 1),

so M is formally equivalent to the real algebraic model M_{N}, or,

(2) For any n ≥ 3, there exists a biholomorphic coordinate transformation of a neighborhood of 0, so that in the new coordinate system, the defining equations of M are

y_{2} = O(n + 1)
z_{3} = z̄_{1}^{2} + O(n + 1)
z_{4} = z_{1}z̄_{1} + O(n + 1),

so M is formally equivalent to the real algebraic model M_{∞}, or,

(3) There exists a biholomorphic coordinate transformation of a neighborhood of 0, so that in the new coordinate system, the defining equations of M are

y_{2} = Q_{2} + O(3)
z_{3} = q_{3} + O(3)
z_{4} = q_{4} + O(3),

where (Q_{2}, q_{3}, q_{4}) is one of the types (III)–(XII) from Section 2.

References

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

E-mail address: CoffmanA@ipfw.edu