CR SINGULARITIES OF REAL FOURFOLDS IN ℂ^{3}

ADAM COFFMAN

Abstract. CR singularities of real 4-submanifolds in complex 3-space are classified by using local holomorphic coordinate changes to transform the quadratic coefficients of the real analytic defining equation into a normal form. The quadratic coefficients determine an intersection index, which appears in global enumerative formulas for CR singularities of compact submanifolds.

1. Introduction

If a real 4-manifold M is embedded in ℂ^{3}, then for each point x on M there are two possibilities: the tangent 4-plane at x is either a complex hyperplane in ℂ^{3}, so M is said to be “CR singular” at x, or it is not, so M is said to be “CR generic” at x. This article considers the local extrinsic geometry of a real analytically embedded M near a CR singular point, by finding invariants under biholomorphic coordinate changes. The main result is a classification of CR singularities, via a list of normal forms for the quadratic part of the local defining function. The matrix algebra leading to the classification is worked out in Section 2, and then summarized in Section 7 after the geometric interpretation is developed in Sections 3 – 5.

The analysis of normal forms near CR singular points is part of the program of studying the local equivalence problem for real m-submanifolds of ℂ^{n}. In this paper we consider the m = 2n − 2 case (“codimension 2”), focusing on real 4-manifolds in ℂ^{3}, since the m = n = 2 case is well-known and larger dimensions seem to lead to difficult computations.

In Section 3, we recall some of Lai’s formulas relating the global topology of a real submanifold to the number of its CR singular points, counted with sign according to an intersection number. In Section 4, we derive a simple expression that calculates the intersection index in terms of the coefficients in the local defining equation, generalizing the well-known m = n = 2 case, where CR singular points of compact surfaces in ℂ^{2} can be counted according to their elliptic or hyperbolic nature as determined by the local Bishop invariant. Section 6 gives some concrete examples of compact real 4-manifolds immersed in ℂ^{3} or ℂP^{3} to illustrate the enumerative formulas and local invariants.

2000 Mathematics Subject Classification. Primary 32V40; Secondary 15A21, 32S05, 32S20.

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

2. Normal forms for CR singularities

Let n ≥ 2 and m = 2n − 2, so a real m-submanifold M of a complex n-manifold has real codimension 2. Considering n in general shows how the well-known (m, n) = (2, 2) case is related to the higher-dimensional cases, including (4, 3). In this Section, we are only interested in a small coordinate neighborhood, so we let the ambient complex space be ℂ^{n}, with coordinates z_{1}, …, z_{n}. We also use the abbreviations z = (z_{1}, …, z_{n−1})^{T} and z⃗ = (z_{1}, …, z_{n})^{T} (both column vectors). The real and imaginary parts of the coordinate functions are labeled z_{k} = x_{k} + iy_{k}.

2.1. Local defining equations and transformations.

We begin by assuming M is a real analytic (2n − 2)-submanifold in ℂ^{n} with a CR singularity at some point: the tangent space at that point is a complex hyperplane. We are interested in the invariants of M under local biholomorphic transformations. By a translation that moves the CR singular point to the origin 0, and then a complex linear transformation of ℂ^{n}, the tangent (2n−2)-plane T0M can be assumed to be the (z_{1}, …, z_{n−1})-subspace. Then there is some neighborhood Δ of the origin in ℂ^{n} so that the defining equation of M in Δ is in the form of a graph over a neighborhood D of the origin in the complex subspace T0M:

z_{n} = h(z_{1}, z̄_{1}, …, z_{n−1}, z̄_{n−1}) = h(z, z̄)

where h(z, z̄) is a complex valued real analytic function defined for z ∈ D ⊆ T0M, and vanishing to second order at z = (0, …, 0)^{T}. Once M is in this “standard position,” the complex defining function h(z, z̄) is of the following form:

(2.1)  h(z, z̄) = z^{T} Qz + z̄^{T} Rz + z̄^{T} Sz̄ + e(z, z̄),

where Q, R, S are complex (n − 1) × (n − 1) coefficient matrices, z^{T} and z̄^{T} are row vectors, and e(z, z̄) is a real analytic function on D vanishing to third order at z = (0, …, 0)^{T}. The matrices Q and S can be assumed to be complex symmetric. It can also be assumed that D is small enough so that the function h(z, z̄) can be expressed as the restriction to {(z, w) ∈ D × D : w_{k} = z̄_{k}} of the multi-indexed series:

(2.2)  h(z, w) = z^{T} Qz + w^{T} Rz + w^{T} Sw + ∑_{|α|+|β|≥3} e^{αβ}z^{α}w^{β},

which converges on some set

(2.3)  D_{c} = {(z, w) : |z_{k}| < ϵ, |w_{k}| < ϵ} ⊆ ℂ^{2n−2}

to a complex analytic function.

Definition 2.1. A (formal, with multi-indexed complex coefficient e^{αβ}) monomial of the form e^{αβ}z^{α}w^{β} = e^{αβ}z_{1}^{α_{1}} ⋯ z_{n−1}^{α_{n−1}}w_{1}^{β_{1}} ⋯ w_{n−1}^{β_{n−1}} has “degree” |α|+|β| = α_{1}+⋯ α_{n−1}+β_{1}+⋯ β_{n−1}. A power series (convergent or formal) e(z, w) = ∑ e^{αβ}z^{α}w^{β} with e^{αβ} = 0 for all terms of degree |α| + |β| < d is abbreviated e(z, w) = O(d).

The above notation applies to expressions of the form e(z, z̄), for example, the last term in Equation (2.1) is e(z, z̄) = O(3).

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

(2.4)  z̃_{1} = z_{1} + p_{1}(z_{1}, …, z_{n})
⋮
z̃_{n} = z_{n} + p_{n}(z_{1}, …, z_{n}),

abbreviated z̃ = z⃗ + p⃗(z⃗), where p_{1}(z⃗), …, p_{n}(z⃗) are holomorphic functions defined by series centered at 0 with no linear or constant terms. Since this transformation of ℂ^{n} has its linear part equal to the identity map, it is invertible on some neighborhood of the origin and preserves the form of (2.1). In the following calculations, we neglect considering the size of that neighborhood, and consider only points close enough to the origin.

As the first special case of a transformation of the form (2.4) to be used, let p_{1}(z⃗), …, p_{n−1}(z⃗) be identically zero, so z̃ = z, and let p_{n}(z⃗) be a homogeneous quadratic polynomial in z_{1}, …, z_{n−1}, so p_{n}(z⃗) = z^{T} Q′z for some complex symmetric (n − 1) × (n − 1) matrix Q′. Given a point on M near 0, its coordinates z⃗ = (z_{1}, …, z_{n})^{T} satisfy z_{n} −h(z, z̄) = 0. The new coordinates at that point satisfy:

(2.5)  z̃_{n} = z_{n} + p_{n}(z⃗)
= z^{T} Qz + z̄^{T} Rz + z̄^{T} Sz̄ + e(z, z̄) + p_{n}(z⃗)
= z̃^{T} (Q + Q′)z̃ + ¯ z̃^{T} Rz̃ + ¯ z̃^{T} S¯z̃ + e(z̃, ¯ z̃).

So, such a quadratic transformation changes the coefficient matrix Q, but all the other coefficients of the new equation, z̃_{n} − h̃(z̃, ¯ z̃) = 0, are the same. Choosing Q′ = S̄−Q (and dropping the tilde notation), the defining equation in the new coordinates is:

(2.6)  z_{n} = z^{T} S̄z + z̄^{T} Rz + z̄^{T} Sz̄ + e(z, z̄),

so the first and third terms have a real valued sum and e(z, z̄) is still O(3).

Next, we consider some linear transformations of ℂ^{n}, but only those which fix, as a set, the complex tangent hyperplane T0M = {z_{n} = 0}, so they are of the form

(2.7) z̃_{n×1} = C_{n×n}z⃗_{n×1} = ( \begin{matrix}c_{1,1} & … & c_{1,n−1} & c_{1,n} \\ ⋮ & & ⋮ & ⋮ \\ c_{n−1,1} & … & c_{n−1,n−1} & c_{n−1,n} \\ 0 & … & 0 & c_{n,n} \end{matrix} ) z⃗_{n×1},

with complex entries and nonzero determinant, so c_{n,n} ≠ 0. The inverse matrix has the block form

C^{−1} = ( \begin{matrix}& & & ∗ \\ & A_{(n−1)×(n−1)} & & ⋮ \\ & & & ∗ \\ 0 & … & 0 & c_{n,n}^{−1} \end{matrix} ) ,

where the block A in C^{−1} does not depend on the entries c_{1,n}, …, c_{n,n}. In the special case where c_{1,n} = … = c_{n−1,n} = 0, C has a block diagonal pattern and so does its inverse, so z = Az̃. In the coordinate system defined by such a linear transformation, the new defining equation is

z̃_{n} = c_{n,n}z_{n}
= c_{n,n} · (z^{T} S̄z + z̄^{T} Rz + z̄^{T} Sz̄ + e(z, z̄))
(2.8)  = c_{n,n} · (z̃^{T} A^{T} S̄Az̃ + ¯ z̃^{T} Ā^{T} RAz̃ + ¯ z̃^{T} Ā^{T} SĀ¯z̃ + ẽ(z̃, ¯ z̃)),

where the new higher order part is still real analytic but may have a different domain of convergence.

If the coefficients c_{1,n}, …, c_{n−1,n} were nonzero, they would contribute only terms of degree at least 3, not affecting the quadratic terms in (2.8). Similarly, allowing a coordinate change with nonlinear terms, as in (2.4), would only introduce terms of degree at least 3, or, as in (2.5), arbitrarily alter the first quadratic term. So, under a general transformation,

(2.9)  z̃ = Cz⃗ + p⃗(z⃗)

(which combines (2.4) and (2.7), and preserves the standard position, (2.1)), the only interesting effect on the quadratic part of h(z, z̄) is that the coefficient matrices are transformed as:

(2.10)  (R, S) ↦ (c_{n,n}Ā^{T} RA, c_{n,n}Ā^{T} SĀ).

The first invariant to notice is the pair (rank(R), rank(S)). The rank of the concatenated matrix (R|S)_{(n−1)×(2n−2)} is also an invariant under this action.

The group of invertible matrices A has (n−1)^{2} complex dimensions, and the group of scalars c_{n,n} is one-dimensional; however, if A is a real multiple λ of the identity matrix 𝟙, then its action can be canceled by choosing c_{n,n} = λ^{−2}. So, there are at most 2((n − 1)^{2} + 1) − 1 = 2n^{2} − 4n + 3 real parameters in the group action. The coefficient matrices R and S have (n − 1)^{2} and (n − 1)n/2 complex dimensions, for a total of 3n^{2} − 5n + 2 real dimensions. The number of coefficients always exceeds the number of parameters in the group action, so we expect infinitely many equivalence classes of matrix pairs, distinguished by continuous invariants.

2.2. Degrees of flatness. We continue with the assumption that M is a real analytic submanifold of ℂ^{n} with real codimension 2.

Definition 2.2. A manifold M in standard position (2.1) has a defining function h(z, z̄) in a “quadratically flat normal form” if the quadratic part, z^{T} Qz + z̄^{T} Rz + z̄^{T} Sz̄, of its defining function is a real valued polynomial. A manifold M ⊆ ℂ^{n} with a CR singular point x⃗ ∈ M is “quadratically flat” at x⃗ if, after M is put into standard position (2.1) by a complex affine transformation z̃ = L_{n×n} · (z⃗ − x⃗), there is a local holomorphic coordinate change (2.9) such that in the new coordinates, M has a defining function in a quadratically flat normal form.

The definition of “quadratically flat normal form” is equivalent to Q = S̄ and R = R̄^{T} (so R is Hermitian symmetric). Considering (2.6) and the transformation rule (2.10), for M in standard position, the “quadratically flat” property is equivalent to R being a complex scalar multiple of a Hermitian symmetric matrix.

The notion of quadratic flatness is the d = 2 special case of the following generalization to higher degree.

Definition 2.3. For d ≥ 2, a real analytic manifold M in standard position (2.1) has a defining function in a “d-flat normal form” if the defining equation in a neighborhood of 0 is

z_{n} = h(z, z̄) = r(z, z̄) + O(d + 1)

for some real valued polynomial r(z, z̄). A manifold M ⊆ ℂ^{n} with a CR singular point x⃗ ∈ M is “d-flat” at x⃗ if, after M is put into standard position (2.1) by a complex affine transformation z̃ = L_{n×n} · (z⃗ − x⃗), there is a local holomorphic coordinate change (2.9) such that in the new coordinates, M has a defining function in a d-flat normal form.

Definition 2.4. A real analytic (2n − 2)-manifold M ⊆ ℂ^{n} is “formally flattenable” at a CR singular point x⃗ ∈ M if it is d-flat at x⃗ for every d ≥ 2.

Definition 2.5. A manifold M ⊆ ℂ^{n} with a CR singular point x⃗ ∈ M is “holomorphically flat” at x⃗ if, after M is put into standard position (2.1) by a complex affine transformation z̃ = L_{n×n} · (z⃗ − x⃗), there is a local holomorphic coordinate change (2.9) such that in the new coordinates, the defining function (2.1) is real valued.

By the Definition, if M is holomorphically flat near a CR singular point, then there is a local coordinate system around the point so that a neighborhood of 0 in M is contained in the real hyperplane {Im(z_{n}) = 0}. By the well-known normal form result of É. Cartan that a real analytic nonsingular Levi flat hypersurface is locally biholomorphically equivalent to a real hyperplane, the local notion of M being holomorphically flat at x⃗ is equivalent to the (more coordinate-free) property that there exists a real analytic nonsingular Levi flat hypersurface containing a neighborhood of x⃗ in M.

2.3. The m = n = 2 case.

When m = n = 2, M is a real surface in ℂ^{2} with a CR singular point. For M in standard position (2.1), the coefficient matrices are size 1 × 1, and can be written as complex constant coefficients. The action of (2.10) becomes (R, S) ↦ (c_{2,2}|α|^{2}R, c_{2,2}ᾱ^{2}S) for nonzero complex constants c_{2,2} and α, where A_{1×1} = (α). If R ≠ 0, then (R, S) can then be transformed into (1, γ_{1}), γ_{1} ≥ 0. If R = 0, then there are two normal forms: (0, 1) and (0, 0). The quadratic normal forms for the defining function of M are then:

(2.11)  z_{2} = z_{1}z̄_{1} + γ_{1} · (z_{1}^{2} + z̄_{1}^{2}) + e(z_{1}, z̄_{1}), γ_{1} ≥ 0, or
z_{2} = z_{1}^{2} + z̄_{1}^{2} + e(z_{1}, z̄_{1}), or
z_{2} = e(z_{1}, z̄_{1}).

So, γ_{1} is the well-known Bishop invariant ([Bishop]) and the second case is γ_{1} = +∞. This calculation of the quadratic normal forms shows that any surface M with a CR singular point is quadratically flat (Definition 2.2) at that point.

The normalization of the cubic terms depends on γ_{1}; all the cases 0 ≤ γ_{1} ≤ ∞ are surveyed in [C_{3}] §5, and we recall a few examples here.

For γ_{1} ∈ (0, \frac{1}{2}) ∪ (\frac{1}{2}, 1) ∪ (1, ∞), it was shown by [MW] that the cubic terms of e(z_{1}, z̄_{1}) can be eliminated by a holomorphic coordinate change near the origin, and so M is 3-flat. (There may be O(4) terms that cannot be eliminated or made real valued.)

Any M with γ_{1} = \frac{1}{2} is 3-flat; although there may be some cubic terms that cannot be eliminated by a holomorphic coordinate change, such terms can always be made real valued. For γ_{1} = 1, there are some M which are not 3-flat.

It was also shown by [MW], and [HK] respectively, that for 0 < γ_{1} < \frac{1}{2}, and γ_{1} = 0, M is holomorphically flat. Results of [MW] and [Gong] show that there exist some surfaces M with γ_{1} > \frac{1}{2} which are formally flattenable but not holomorphically flat.

2.4. The m = 4, n = 3 case.

2.4.1. A quadratic normal form. In this, the main case of this paper, M is a 4-manifold in ℂ^{3}, which we assume is given in the form (2.6). The quadratic coefficient matrices R, S are size 2 × 2, so there are 7 independent complex coefficients, and, as previously calculated, 9 real parameters in the group action (2.10). One expects that in general, attempting to put the pair (R, S) into a normal form leaves 5 continuous real invariants.

We choose to begin the normalization by considering the action of the transformation R_{2×2} ↦ cĀ^{T} RA, where c = c_{3,3} is a nonzero scalar and A is an invertible 2 × 2 complex matrix. Conveniently, the problem of finding representative matrices for the orbits of this action has already been solved in [C_{2}]:

Proposition 2.6 ([C_{2}] Theorem 4.3). Given a complex 2 × 2 matrix R, there is exactly one of the following normal forms N such that N = cĀ^{T} RA for some nonzero complex c and some invertible complex A_{2×2}:

(1) ( \begin{matrix}1 & 0 \\ 0 & e^{iθ} \end{matrix} ), 0 ≤ θ ≤ π;

(2) ( \begin{matrix}0 & 1 \\ τ & 0 \end{matrix} ), 0 ≤ τ < 1;

(3) ( \begin{matrix}0 & 1 \\ 1 & i \end{matrix} );

(4) ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} );

(5) ( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} ).

For most values of the invariants θ, τ, these normal forms are not Hermitian symmetric — unless R is already a complex multiple of a Hermitian matrix, one would not expect cĀ^{T} RA to be Hermitian. So, unlike the m = n = 2 case from Subsection 2.3, the quadratic part of the defining function h(z, z̄) generally cannot be made real valued by a holomorphic coordinate change. For a manifold in standard position (2.1), the following are equivalent: (I) M is quadratically flat at 0; (II) R is a multiple of a Hermitian matrix; (III) N = cĀ^{T} RA, where N is one of the following normal forms from the Proposition: Case (1) with θ = 0 or π, Case (4), or Case (5).

Using N from the above Proposition and introducing the notation P = 2S̄_{2×2}, (2.6) becomes:

(2.12) z_{3} = (z̄_{1}, z̄_{2})N ( \begin{matrix}z_{1} \\ z_{2} \end{matrix} ) + Re ((z_{1}, z_{2})P ( \begin{matrix}z_{1} \\ z_{2} \end{matrix} )) + e(z_{1}, z̄_{1}, z_{2}, z̄_{2}).

The linear action of C (2.7, 2.8), followed by another holomorphic transformation (2.5), preserves the form of the defining equation (2.12), and acts on the coefficient matrices by the transformation:

(2.13)  (N, P) ↦ (cĀ^{T} NA, c̄A^{T} P A).

Since N is already normalized, to find a normal form for (2.12), we consider only pairs (c, A) that preserve N: N = cĀ^{T} NA. Since this depends on the nature of the various matrices N appearing in the Proposition, we proceed in cases.

In each case, let P = ( \begin{matrix}a & b \\ b & d \end{matrix} ), with complex entries a, b, d, and let

A = ( \begin{matrix}α & β \\ γ & δ \end{matrix} ), with complex entries and nonzero determinant.

Case (1a). For N = ( \begin{matrix}1 & 0 \\ 0 & e^{iθ} \end{matrix} ), with 0 < θ < π, if N = cĀ^{T} NA, then c

is real (this follows from calculating determinants, for example), and

N = cĀ^{T} NA = c ( \begin{matrix}ᾱ & γ̄ \\ β̄ & δ̄ \end{matrix} ) ( \begin{matrix}1 & 0 \\ 0 & e^{iθ} \end{matrix} ) ( \begin{matrix}α & β \\ γ & δ \end{matrix} )

implies

(2.14)  1 = c · (αᾱ + γγ̄e^{iθ}),
0 = αβ̄ + γδ̄e^{iθ},
e^{iθ} = c · (ββ̄ + δδ̄e^{iθ}).

It follows that γ = β = 0, c = |α|^{−2}, and |δ| = |α|. So the action of (2.13) is that P can be transformed to:

c̄A^{T} P A = \frac{1}{|α|^{2}} ( \begin{matrix}α & 0 \\ 0 & δ \end{matrix} ) ( \begin{matrix}a & b \\ b & d \end{matrix} ) ( \begin{matrix}α & 0 \\ 0 & δ \end{matrix} ) = ( \begin{matrix}a\frac{α^{2}}{|α|^{2}} & b\frac{αδ}{|αδ|} \\ b\frac{αδ}{|αδ|} & d\frac{δ^{2}}{|δ|^{2}} \end{matrix} ) .

When a and d are both nonzero, α and δ can be chosen to rotate them onto the positive real axis. The value of b cannot be normalized any further except that P with positive a, d, and complex b is equivalent to the matrix with the same a, d, but opposite value for b.

If a = 0 or d = 0, then α and δ can be chosen to transform P into a matrix with all nonnegative entries.

Case (1b). For N = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ), the θ = 0 case of Proposition 2.6, c is

real as in the previous case, and by the calculation analogous to (2.14) with θ = 0, in fact c is positive, so c = \frac{+1}{| det(A)|}. The equation N = cĀ^{T} NA =

\overline{(\sqrt{c}A)}^{T} N(\sqrt{c}A) shows A is a real multiple of a unitary matrix, and conversely if A = rU for some real r and unitary U, then (c, A) = (r^{−2}, rU) stabilizes N. So the action of (2.13) is that P can be transformed to:

c̄A^{T} P A = \frac{1}{| det(A)|}A^{T} P A = r^{−2}(rU)^{T} P (rU) = U^{T} P U.

The normal form problem for P is thus reduced to finding a normal form for a complex symmetric 2 × 2 matrix under the relation of congruence by a unitary matrix. This problem has a well-known solution by Takagi ([HJ] §4.4, see also Theorem 5 of [Hua]), which says that a complex symmetric matrix has a diagonal normal form under unitary congruence, with nonnegative real entries. These entries can be re-ordered by a unitary transformation, so a

normal form is ( \begin{matrix}a & 0 \\ 0 & d \end{matrix} ) with 0 ≤ a ≤ d.

Case (1c). For N = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ), the θ = π case of Proposition 2.6, c

is real. Briefly neglecting c, we consider the group of invertible matrices A preserving N. The condition Ā^{T} NA = N is equivalent to ANĀ^{T} = N. For any symmetric coefficient matrix P, define an auxiliary matrix B = P̄^{T} NP, which is Hermitian symmetric. Then the congruence action P′ = A^{T} P A transforms the product:

B′ = \overline{P′}^{T} NP′ = \overline{(A^{T} P A)}^{T} N(A^{T} P A) = Ā^{T} P̄^{T} ĀNA^{T} P A = Ā^{T} P̄^{T} NP A,

so B′ is related to B by Hermitian congruence. So, the action (2.13) of A on (N, P) (with c = 1) has been temporarily replaced by the action of simultaneous Hermitian congruence on the pair (N, B) of Hermitian symmetric matrices. This normal form problem is considered by [HJ], and more recently by [HS] and [LR]. Recalling that N is Hermitian congruent to the matrix

N′ = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) (the τ → 1^{−} limit of Case (2) from the Proposition), the

result from [HJ], [HS], [LR] is that for any Hermitian B, the pair (N, B) is equivalent under simultaneous Hermitian congruence to exactly one pair from the following list:

• (( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) , ( \begin{matrix}k_{1} & 0 \\ 0 & k_{2} \end{matrix} )), k_{1}, k_{2} ∈ ℝ;

• (( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) , ( \begin{matrix}0 & k \\ k & 1 \end{matrix} )), k ∈ ℝ;

• (( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) , ( \begin{matrix}0 & x + iy \\ x − iy & 0 \end{matrix} )), x ∈ ℝ, y > 0.

We continue with Case (1c) by splitting into subcases corresponding to the above three intermediate normal forms.

Case (1ci). There is some nonsingular matrix A so that the Hermitian pair (N, P̄^{T} NP) is simultaneously diagonalized and P′ = A^{T} P A is a complex

symmetric matrix ( \begin{matrix}a & b \\ b & d \end{matrix} ) (using the same place-holding letters as in Case

(1a) even though P has been transformed once already) satisfying

(2.15) B′ = \overline{P′}^{T} NP′ = \overline{( \begin{matrix}a & b \\ b & d \end{matrix} )}^{T} ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) ( \begin{matrix}a & b \\ b & d \end{matrix} ) = ( \begin{matrix}k_{1} & 0 \\ 0 & k_{2} \end{matrix} ) .

By another transformation, of the form e^{iξ}𝟙 (which does not affect the pair (N, B′)), we may assume that the entry b of P′ satisfies b ≥ 0. It then follows from expanding (2.15) that the entries of P′ must satisfy either b = 0 or d = ā.

In the b = 0 case, a transformation of the form ( \begin{matrix}α & 0 \\ 0 & δ \end{matrix} ), with |α| = |δ| = 1,

preserves N and puts P′ into a diagonal normal form with nonnegative real

entries. By a transformation of the form (c, A) = (−1, ( \begin{matrix}0 & i \\ i & 0 \end{matrix} )), these

entries can be interchanged, so a unique normal form is ( \begin{matrix}a & 0 \\ 0 & d \end{matrix} ) with 0 ≤

a ≤ d.

In the d = ā case, the same type of diagonal transformation with δ = 1/α

puts P′ into the form ( \begin{matrix}a & b \\ b & a \end{matrix} ) with b > 0 and a ≥ 0. However, this

simplifies even further, and is not always different from the previous (b = 0) case. When 0 < b < a, a transformation of the form

(c, A) = (\frac{−1}{2\sqrt{a^{2} − b^{2}}(−a + \sqrt{a^{2} − b^{2}})}, ( \begin{matrix}b & −a + \sqrt{a^{2} − b^{2}} \\ −a + \sqrt{a^{2} − b^{2}} & b \end{matrix} ))

preserves N and diagonalizes P′ to \sqrt{a^{2} − b^{2}} · 𝟙. When 0 ≤ a < b, N and P′ cannot be simultaneously diagonalized, but a transformation of the form

(c, A) = (\frac{1}{4b\sqrt{b^{2} − a^{2}}} , ( \begin{matrix}−a + i(b + \sqrt{b^{2} − a^{2}}) & b − \sqrt{b^{2} − a^{2}} − ia \\ b − \sqrt{b^{2} − a^{2}} − ia & −a + i(b + \sqrt{b^{2} − x^{2}}) \end{matrix} ))

preserves N and takes P′ to \frac{−i}{b} \sqrt{b^{2} − a^{2}}(a − i\sqrt{b^{2} − a^{2}})N′, which can be rotated by A = e^{iθ}𝟙 to \sqrt{b^{2} − a^{2}}N′. In the 0 < a = b case, P′ has rank 1 but (N, P′) is not simultaneously diagonalizable, so it is inequivalent to the b = 0 case with rank 1, where 0 = a < d. A transformation of the form

(c, A) = (\frac{1}{4a}, ( \begin{matrix}1 + a & 1 − a \\ 1 − a & 1 + a \end{matrix} ))

preserves N and normalizes a to 1.

Case (1cii). If there is no transformation simultaneously diagonalizing (N, B), then there is some nonsingular matrix A so that Ā^{T} NA = N′, A^{T} P A = P′, and B′ = Ā^{T} BA equals either the second or third normal form from the above list — in this subcase we consider the second. The property Ā^{T} NA = N′ is equivalent to ĀN′A^{T} = N, so

B′ = Ā^{T} BA = Ā^{T} P̄^{T} NP A = Ā^{T} P̄^{T} ĀN′A^{T} P A = \overline{(A^{T} P A)}^{T} N′(A^{T} P A).

With notation as in the previous case, P′ = A^{T} P A is a complex symmetric

matrix ( \begin{matrix}a & b \\ b & d \end{matrix} ) satisfying

(2.16) B′ = \overline{P′}^{T} N′P′ = \overline{( \begin{matrix}a & b \\ b & d \end{matrix} )}^{T} ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) ( \begin{matrix}a & b \\ b & d \end{matrix} ) = ( \begin{matrix}0 & k \\ k & 1 \end{matrix} ) .

By another transformation, of the form e^{iξ}𝟙 (which does not affect the pair (N′, B′)), we may assume that the entry a of P′ satisfies a ≥ 0. However, it follows from expanding the product in (2.16) that there are no solutions of (2.16) with a > 0, so a = 0. Then (2.16) becomes

\overline{( \begin{matrix}0 & b \\ b & d \end{matrix} )}^{T} ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) ( \begin{matrix}0 & b \\ b & d \end{matrix} ) = ( \begin{matrix}0 & bb̄ \\ bb̄ & bd̄+ b̄d \end{matrix} ) = ( \begin{matrix}0 & k \\ k & 1 \end{matrix} ) ,

so neither b nor d is 0. By yet another scalar transformation, we may assume that b > 0, without changing the RHS of the above equation, so b is an invariant determined by k, and the equality of entries b · (d + d̄) = 1 implies

Re(d) ≠ 0. A transformation (c, A) = (\frac{1}{r}, ( \begin{matrix}1 & is \\ 0 & r \end{matrix} )), with r and s real,

preserves N′ and transforms P′ = ( \begin{matrix}0 & b \\ b & d \end{matrix} ) into ( \begin{matrix}0 & b \\ b & 2bis + rd \end{matrix} ). Since

Re(d) ≠ 0 and b > 0, r and s can be chosen to normalize d to 1.

Case (1ciii). The third case starts with the same steps as the previous case,

with Ā^{T} NA = N′ and A^{T} P A = P′ = ( \begin{matrix}a & b \\ b & d \end{matrix} ) satisfying

(2.17)

B′ = \overline{P′}^{T} N′P′ = \overline{( \begin{matrix}a & b \\ b & d \end{matrix} )}^{T} ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) ( \begin{matrix}a & b \\ b & d \end{matrix} ) = ( \begin{matrix}0 & x + iy \\ x − iy & 0 \end{matrix} ) ,

with y > 0. By another transformation, of the form e^{iξ}𝟙 (which does not affect the pair (N′, B′)), we may assume that the entry a of P′ satisfies a ≥ 0. Then it follows from expanding the product in (2.17) that a > 0, b = 0, and ad = x+

iy, an invariant quantity. A transformation (c, A) = (a, ( \begin{matrix}1/a & 0 \\ 0 & 1 \end{matrix} )) pre-

serves N′ and transforms P′ = ( \begin{matrix}a & 0 \\ 0 & d \end{matrix} ) into ( \begin{matrix}1 & 0 \\ 0 & ad \end{matrix} ) = ( \begin{matrix}1 & 0 \\ 0 & x + iy \end{matrix} ).

Case (2a). For N = ( \begin{matrix}0 & 1 \\ τ & 0 \end{matrix} ), with 0 < τ < 1, if N = cĀ^{T} NA, then c is

real, and

N = cĀ^{T} NA = c ( \begin{matrix}ᾱ & γ̄ \\ β̄ & δ̄ \end{matrix} ) ( \begin{matrix}0 & 1 \\ τ & 0 \end{matrix} ) ( \begin{matrix}α & β \\ γ & δ \end{matrix} )

implies

(2.18)  0 = αγ̄τ + γᾱ,
0 = βδ̄τ + β̄δ,
1 = c · (βγ̄τ + ᾱδ),
τ = c · (γβ̄ + αδ̄τ).

It follows that γ = β = 0, ᾱδ is real, and c = (ᾱδ)^{−1}. So the action of (2.13) is that P can be transformed to:

c̄A^{T} P A = \frac{1}{ᾱδ} ( \begin{matrix}α & 0 \\ 0 & δ \end{matrix} ) ( \begin{matrix}a & b \\ b & d \end{matrix} ) ( \begin{matrix}α & 0 \\ 0 & δ \end{matrix} ) = ( \begin{matrix}aα/δ̄ & bα/ᾱ \\ bα/ᾱ & dδ/ᾱ \end{matrix} ) .

When b ≠ 0, α can be chosen to rotate it onto the positive real axis. Then, using real α and δ, a can be either scaled onto the unit circle or the origin; if a = 0, then d can be scaled onto the unit circle or the origin. The resulting normal form is unique except that (a, d) is equivalent to the pair (−a, −d).

If b = 0 and a ≠ 0, then α and δ can be chosen to transform a into 1, leaving d ∈ ℂ as an invariant. If b = a = 0, α and δ can be chosen to transform d into 1 or 0.

Case (2b). For N = ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ), the τ = 0 case of Proposition 2.6, if

N = cĀ^{T} NA, then c does not have to be real, but the calculation is similar to the previous (2a) case. The τ = 0 analogue of (2.18) implies γ = β = 0 and c = (ᾱδ)^{−1}. So the action of (2.13) is that P can be transformed to:

c̄A^{T} P A = (αδ̄)^{−1} ( \begin{matrix}α & 0 \\ 0 & δ \end{matrix} ) ( \begin{matrix}a & b \\ b & d \end{matrix} ) ( \begin{matrix}α & 0 \\ 0 & δ \end{matrix} ) = δ̄^{−1} ( \begin{matrix}aα & bδ \\ bδ & d\frac{δ^{2}}{α} \end{matrix} ) .

When b ≠ 0, δ can be chosen to rotate it onto the positive real axis. Then, using real δ and a complex number α, d can be transformed to 1, leaving a ∈ ℂ as an invariant, or to 0 and then a can be transformed to 1 or 0.

If b = 0 and d ≠ 0, then α and δ can be chosen to transform d into 1 and a to a nonnegative invariant. If b = d = 0, α and δ can be chosen to transform a into 1 or 0.

Case (3). For N = ( \begin{matrix}0 & 1 \\ 1 & i \end{matrix} ), the group of (c, ( \begin{matrix}α & β \\ γ & δ \end{matrix} )) such that

N = cĀ^{T} NA is exactly the set with γ = 0, α = δ ≠ 0, αβ̄ + ᾱβ = 0, and c|α|^{2} = 1. Then

c̄A^{T} P A = \frac{1}{|α|^{2}} ( \begin{matrix}aα^{2} & aαβ + bα^{2} \\ aαβ + bα^{2} & aβ^{2} + 2bαβ + dα^{2} \end{matrix} ) .

If a ≠ 0, then β can eliminate b, α can rotate a to the positive real axis, and the complex number in the d position is an invariant. If a = 0 and b ≠ 0, then β can eliminate d and α can rotate b to the positive real axis. If a = b = 0, then α can rotate d to the nonnegative real axis.

Case (4). For N = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ), the group of (c, ( \begin{matrix}α & β \\ γ & δ \end{matrix} )) such that

N = cĀ^{T} NA is exactly the set with β = 0, δ ≠ 0, and c|α|^{2} = 1. The entry d of P can be normalized to 1 or 0. In the d = 1 case, γ can eliminate any b entry, and α can rotate a onto the nonnegative real axis. In the d = 0 case, if b ≠ 0, then (a, b) can be normalized to (0, 1); if b = 0, then α can rotate a onto the nonnegative real axis.

Case (5). Under an arbitrary congruence transformation, the rank is the only invariant of a complex symmetric matrix P under a congruence transformation A^{T} P A. The three normal forms are:

• P = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} );

• P = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} );

• P = ( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} ).

All these results on normal forms for the matrix pairs (N, P) are summarized by Table 1 in Section 7.

2.4.2. An alternative quadratic normal form. Instead of choosing to normalize the R matrix first in (2.6), we could have chosen to normalize the symmetric matrix S. Then the calculations start off in a simpler way, since first applying the congruence transformation to the complex symmetric matrix S, the three normal forms for Ā^{T} SĀ are exactly as in the above Case (5).

In the first case, S = 𝟙, the normalization problem for (R, S) reduces to the problem of finding a normal form for R under the action R ↦ cĀ^{T} RA, where (c, A) satisfies cĀ^{T} Ā = 𝟙. This is the generic form of S, so one still expects five continuous real parameters in any collection of representative matrix pairs, but we do not attempt to find such normal forms.

In the second case, the pair (c, A) stabilizing S and acting on R satisfies

cĀ^{T} ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) Ā = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) ,

leading to another normal form problem for R which we again do not pursue.

In the last case, where S is the zero matrix, the R matrix can then be put into one of the normal forms N from Proposition 2.6.

For the special case where R is Hermitian, [I] gives a list of 2 × 2 normal forms for (R, S), following this approach of normalizing S first.

2.4.3. One example of a cubic normal form. With M in standard position and the quadratic part of the defining function in normal form (2.12), we can consider its cubic terms. In the expansion

z_{3} = (z̄_{1}, z̄_{2})N ( \begin{matrix}z_{1} \\ z_{2} \end{matrix} ) + Re ((z_{1}, z_{2})P ( \begin{matrix}z_{1} \\ z_{2} \end{matrix} )) + e_{3}(z, z̄) + e(z, z̄),

e_{3}(z, z̄) is the cubic part, and e(z, z̄) = O(4). In particular,

(2.19) e_{3} = e^{3000}z_{1}^{3} + e^{2100}z_{1}^{2}z̄_{1} + e^{1200}z_{1}z̄_{1}^{2} + e^{0300}z̄_{1}^{3}
+e^{2010}z_{1}^{2}z_{2} + e^{1110}z_{1}z̄_{1}z_{2} + e^{0210}z̄_{1}^{2}z_{2} + e^{2001}z_{1}^{2}z̄_{2}
+e^{1101}z_{1}z̄_{1}z̄_{2} + e^{0201}z̄_{1}^{2}z̄_{2} + e^{1020}z_{1}z_{2}^{2} + e^{0120}z̄_{1}z_{2}^{2}
+e^{1011}z_{1}z_{2}z̄_{2} + e^{0111}z̄_{1}z_{2}z̄_{2} + e^{1002}z_{1}z̄_{2}^{2} + e^{0102}z̄_{1}z̄_{2}^{2}
+e^{0030}z_{2}^{3} + e^{0021}z_{2}^{2}z̄_{2} + e^{0012}z_{2}z̄_{2}^{2} + e^{0003}z̄_{2}^{3}.

The holomorphic coordinate changes that fix the origin and preserve the standard position of M are of the form

(2.20) z̃_{1} = c_{11}z_{1} + c_{12}z_{2} + c_{13}z_{3} + p_{1}^{20}z_{1}^{2} + p_{1}^{11}z_{1}z_{2} + p_{1}^{02}z_{2}^{2}
z̃_{2} = c_{21}z_{1} + c_{22}z_{2} + c_{23}z_{3} + p_{2}^{20}z_{1}^{2} + p_{2}^{11}z_{1}z_{2} + p_{2}^{02}z_{2}^{2}
z̃_{3} = c_{33}z_{3} + p_{3}^{200}z_{1}^{2} + p_{3}^{110}z_{1}z_{2} + p_{3}^{020}z_{2}^{2} + p_{3}^{101}z_{1}z_{3} + p_{3}^{011}z_{2}z_{3}
+p_{3}^{300}z_{1}^{3} + p_{3}^{210}z_{1}^{2}z_{2} + p_{3}^{120}z_{1}z_{2}^{2} + p_{3}^{030}z_{2}^{3}.

The linear coefficients are as in (2.7); the p coefficients are from (2.4). Assigning to each monomial z_{1}^{α_{1}}z_{2}^{α_{2}}z_{3}^{α_{3}} a “weight” α_{1} + α_{2} + 2α_{3}, including terms in (2.20) of higher weight, such as z_{3}^{2}, would not contribute any changes to the quadratic or cubic coefficients of the defining function in the z̃ coordinates. The effect of a coordinate change (2.20) on the cubic part e_{3} depends on the quadratic coefficients from N and P. If the quadratic part has already been put into a normal form, then in attempting to find a normal form for the cubic terms, one would want to use only holomorphic transformations that preserve the quadratic normal form, and this subgroup of transformations also depends on the coefficient matrices N and P. The comprehensive problem of finding cubic normal forms for every equivalence class of CR singularities seems to be difficult, so we consider just one special but interesting case.

Let M be given by this defining equation in standard position and with quadratic normal form as in Case (1b) from Subsection 2.4.1:

(2.21) z_{3} = z_{1}z̄_{1} + γ_{1} · (z_{1}^{2} + z̄_{1}^{2}) + z_{2}z̄_{2} + γ_{2} · (z_{2}^{2} + z̄_{2}^{2}) + e_{3}(z, z̄) + e(z, z̄).

The above expression is in a quadratically flat normal form: the quadratic part is real valued with 0 ≤ γ_{1} ≤ γ_{2}, and e_{3} is as above. Now, assume further that 0 < γ_{1} < γ_{2}, with neither γ_{1} nor γ_{2} equal to \frac{1}{2} or 1.

It follows from the normal form result of [MW] (as in Subsection 2.3) that there is a transformation of the form

(2.22)  z̃_{1} = z_{1} + c_{13}z_{3} + p_{1}^{20}z_{1}^{2}
z̃_{2} = z_{2}
z̃_{3} = z_{3} + p_{3}^{101}z_{1}z_{3} + p_{3}^{300}z_{1}^{3}

that eliminates the terms e^{3000}z_{1}^{3} + e^{2100}z_{1}^{2}z̄_{1} + e^{1200}z_{1}z̄_{1}^{2} + e^{0300}z̄_{1}^{3} from e_{3}, without changing the quadratic part. Similarly, a transformation of z_{2}, z_{3} can eliminate the cubic terms depending only on z_{2}, z̄_{2}, without changing the quadratic part or re-introducing the cubic terms in z_{1}, z̄_{1}. This leaves twelve monomials in (2.19) with complex coefficients.

Among the transformations (2.20), the subgroup preserving this partial normal form is given by:

(2.23)  z̃_{1} = c_{11}z_{1} + p_{1}^{11}z_{1}z_{2} + p_{1}^{02}z_{2}^{2}
z̃_{2} = c_{22}z_{2} + p_{2}^{20}z_{1}^{2} + p_{2}^{11}z_{1}z_{2}
z̃_{3} = c_{33}z_{3} + p_{3}^{210}z_{1}^{2}z_{2} + p_{3}^{120}z_{1}z_{2}^{2},

again, omitting terms of higher weight. The linear coefficients c_{11}, c_{22}, c_{33} must be real, and satisfy c_{33} = c_{11}^{2} = c_{22}^{2}. The six complex coefficients of p⃗ can be arbitrary, and then after a coordinate change of the form (2.23), the defining equation becomes:

z̃_{3} = z̃_{1} ¯z̃_{1} + γ_{1} · (z̃_{1}^{2} + ¯ z̃_{1}^{2}) + z̃_{2} ¯z̃_{2} + γ_{2} · (z̃_{2}^{2} + ¯ z̃_{2}^{2}) + ẽ_{3}(z̃, ¯ z̃) + e(z̃, ¯ z̃),
ẽ_{3} = (\frac{e^{1110}}{c_{22}} − \frac{p_{1}^{11}}{c_{11}c_{22}} ) z̃_{1} ¯z̃_{1}z̃_{2} + (\frac{e^{1101}}{c_{22}} − \frac{\overline{p_{1}^{11}}}{c_{11}c_{22}} ) z̃_{1} ¯z̃_{1} ¯z̃_{2}

+ (\frac{e^{2001}}{c_{22}} − \frac{p_{2}^{20}}{c_{11}^{2}} ) z̃_{1}^{2} ¯z̃_{2} + (\frac{e^{0210}}{c_{22}} − \frac{\overline{p_{2}^{20}}}{c_{11}^{2}} ) ¯ z̃_{1}^{2}z̃_{2} (2.24)

+ (\frac{e^{1011}}{c_{11}} − \frac{p_{2}^{11}}{c_{11}c_{22}} ) z̃_{1}z̃_{2} ¯z̃_{2} + (\frac{e^{0111}}{c_{11}} − \frac{\overline{p_{2}^{11}}}{c_{11}c_{22}} ) ¯ z̃_{1}z̃_{2} ¯z̃_{2}

+ (\frac{e^{1002}}{c_{11}} − \frac{\overline{p_{1}^{02}}}{c_{22}^{2}} ) z̃_{1} ¯z̃_{2}^{2} + (\frac{e^{0120}}{c_{11}} − \frac{p_{1}^{02}}{c_{22}^{2}} ) ¯ z̃_{1}z̃_{2}^{2}

+ (\frac{e^{2010}}{c_{22}} − \frac{2γ_{1}p_{1}^{11}}{c_{11}c_{22}} − \frac{2γ_{2}p_{2}^{20}}{c_{11}^{2}} + \frac{p_{3}^{210}}{c_{11}^{2}c_{22}} ) z̃_{1}^{2}z̃_{2}

+ (\frac{e^{0201}}{c_{22}} − \frac{2γ_{1}\overline{p_{1}^{11}}}{c_{11}c_{22}} − \frac{2γ_{2}p_{2}^{20}}{c_{11}^{2}} ) ¯ z̃_{1}^{2} ¯z̃_{2}

+ (\frac{e^{1020}}{c_{11}} − \frac{2γ_{1}p_{1}^{02}}{c_{22}^{2}} − \frac{2γ_{2}p_{2}^{11}}{c_{11}c_{22}} + \frac{p_{3}^{120}}{c_{11}c_{22}^{2}} ) z̃_{1}z̃_{2}^{2}

+ (\frac{e^{0102}}{c_{11}} − \frac{2γ_{1}\overline{p_{1}^{02}}}{c_{22}^{2}} − \frac{2γ_{2}\overline{p_{2}^{11}}}{c_{11}c_{22}} ) ¯ z̃_{1} ¯z̃_{2}^{2}.

In general, there are not enough parameters in (2.23) to put these twelve coefficients into a sparse normal form. We turn to yet a further special case.

Suppose that after M has been partially normalized as in (2.22), so that

e^{3000} = e^{2100} = e^{1200} = e^{0300} = e^{0030} = e^{0021} = e^{0012} = e^{0003} = 0, eight of the remaining twelve coefficients satisfy the following conditions:

(2.25)  e^{1110} = \overline{e^{1101}}
e^{2001} = \overline{e^{0210}}
e^{1011} = \overline{e^{0111}}
e^{1002} = \overline{e^{0120}}.

This condition holds if (but not only if) the partially normalized e_{3} is real valued, so the defining function of M is in a 3-flat normal form. Then, by inspection of (2.24), there is a transformation with complex coefficients p_{1}^{02}, p_{1}^{11}, p_{2}^{11}, p_{2}^{20} that can normalize the all the e^{1110}, …, coefficients in (2.25) to 0. A transformation using p_{3}^{120} and p_{3}^{210} can then change the e^{2010} and e^{1020} coefficients to any value, in particular, to the complex conjugates of e^{0201} and e^{0102}, so that e_{3} can be brought to the following real valued normal form:

(2.26)  ẽ_{3} = \overline{e^{0201}}z̃_{1}^{2}z̃_{2} + e^{0201} ¯z̃_{1}^{2} ¯z̃_{2} + \overline{e^{0102}}z̃_{1}z̃_{2}^{2} + e^{0102} ¯z̃_{1} ¯z̃_{2}^{2}.

The coefficients e^{0201}, e^{0102} are not invariants since a real linear re-scaling by c_{11} and c_{22} = ±c_{11} is still possible.

There is a different, more useful, statement about the conditions under which the above normal form can be achieved:

Theorem 2.7. Suppose the real 4-manifold M in ℂ^{3} is in standard position with defining equation in the quadratically flat normal form (2.21) with 0 < γ_{1} < γ_{2}, and neither γ_{1} nor γ_{2} equal to \frac{1}{2}, and with cubic terms e_{3} as in (2.19) that satisfy the conditions

(2.27)  e^{1200} = \overline{e^{2100}}
(2.28)  e^{0021} = \overline{e^{0012}}
e^{1110} = \overline{e^{1101}}
e^{2001} = \overline{e^{0210}}
e^{1011} = \overline{e^{0111}}
e^{1002} = \overline{e^{0120}}.

Then there is a holomorphic coordinate change (2.20) that puts the cubic part into the 3-flat normal form ẽ_{3} (2.26).

Proof. The last four out of the above six conditions are copied from (2.25). The proof of the Theorem is to proceed as above, first eliminating the cubic terms in z_{1}, z̄_{1} only. The condition (2.27) allows this to be done by a transformation of the form (2.22), but with p_{3}^{101} = 0. (This is possible even in the γ_{1} = 1 case, cf. Example 5.6 of [C_{3}].) Similarly, the cubic terms in z_{2}, z̄_{2} only can be eliminated by a transformation of the form

z̃_{1} = z_{1}
z̃_{2} = z_{2} + c_{23}z_{3} + p_{2}^{02}z_{2}^{2}
(2.29)  z̃_{3} = z_{3} + p_{3}^{030}z_{2}^{3},

without re-introducing any of the previously eliminated terms. Again, the condition (2.28) (and γ_{2} ≠ \frac{1}{2}) means that a term p_{3}^{011}z_{2}z_{3} is not needed in (2.29). These first two steps may alter the numerical values of the remaining coefficients, but the claim is that if the cubic coefficients e^{1110}, …, e^{0120} satisfy the reality conditions (2.25) at the start of the process, then the new corresponding coefficients continue to satisfy those conditions. The calculation to verify this is straightforward but omitted; however, the assumption p_{3}^{101} = p_{3}^{011} = 0 is crucial: if either were nonzero, cubic terms not satisfying (2.25) could appear in the new defining equation. The rest of the normalization proceeds exactly as above.

The conclusion of the above Theorem holds in particular when M has the specified quadratic normal form and is also 3-flat.

2.5. Flatness in higher dimensions.

Proposition 2.6 showed that for the m = 4, n = 3 case considered in Subsection 2.4, R cannot, in general, be put into a Hermitian normal form. Similarly for higher dimensions n > 2, m = 2n−2, quadratic flatness is a nongeneric property for codimension 2 manifolds with CR singularities. Higher degree flatness is even more non-generic.

Consider the quadratically flat case, where M is in standard position in ℂ^{n} and the coefficient matrix R happens to be Hermitian symmetric. This property of R is preserved by the action of (c_{n,n}, A) from (2.7) if and only if c_{n,n} is real — unless R is the zero matrix, in which case there is no such condition on c_{n,n}.

For a nonzero Hermitian matrix R, the transformations (2.10) (or (2.13)) of the defining function that preserve the property of being in a quadratically flat normal form have the following action: R ↦ c_{n,n}Ā^{T} RA, a congruence transformation of R, followed by real scalar multiplication. The Hermitian property of R is preserved, along with its rank. Congruence transformations also preserve the signature (p, q): the number of positive, and negative, eigenvalues; R is “definite” if p = n or q = n. Multiplying by a negative scalar c_{n,n} interchanges the p and q quantities. The rank ρ(R), the number σ(R) = |p−q| (from which p and q can be recovered, modulo switching), and the property of definiteness (or indefiniteness) are invariants of R under the action of this transformation group.

If R is the zero matrix, then c_{n,n}R is still Hermitian even if c_{n,n} is not a real number. M is quadratically flat: the quadratic part of h(z, z̄) can be made real valued by a transformation of the form (2.5).

Including both cases R ≠ 0, R = 0, we can conclude that if M has a defining function h(z, z̄) in a quadratically flat normal form, then any holomorphic coordinate change that preserves the property that h(z, z̄) is in a quadratically flat normal form must leave invariant the rank ρ(R) and the number σ(R) = |p − q|. In particular, these quantities are also invariants of a defining function in a d-flat normal form, or which is real valued (the holomorphically flat case), under holomorphic coordinate changes that preserve the flatness property of the defining function. The determinant of R is real and transforms as: det(c_{n,n}Ā^{T} RA) = c_{n,n}^{n−1}| det(A)|^{2} det(R), so the sign of det(R) is also an invariant if n is odd.

For arbitrary dimensions m = 2n − 2, n ≥ 2, if the (n − 1) × (n − 1) coefficient matrix R in (2.6) is Hermitian and definite (so that R transforms by congruence to the identity matrix 𝟙), the result of Takagi is that then there is a unitary coordinate change diagonalizing S, with real entries 0 ≤ γ_{1} ≤ … ≤ γ_{n−1} on the diagonal. These numbers are called “generalized Bishop invariants” by [HY], since the 0 ≤ γ_{1} < ∞ normal form (2.11) from Subsection 2.3 is the n = 2 special case. The definite normal form from Case (1b) of Subsection 2.4.1 is the n = 3 case. The R = 𝟙, S = 0_{(n−1)×(n−1)} case, where z_{n} = z_{1}z̄_{1} + … + z_{n−1}z̄_{n−1} + O(3), is considered by [HY].

For P complex symmetric and R Hermitian symmetric, but not necessarily definite, a description of canonical representatives for the equivalence relation (R, P) ∼ (Ā^{T} RA, A^{T} P A) is given by [E]. Allowing scalar multiplication, as in (2.13), with real c_{n,n} as above, is different only in that (R, P) is equivalent to (−R, −P) under the action of (2.13), while they may be inequivalent according to [E]. In the 2 × 2 case, the choices made by [I] and [E] for canonical forms are different from those in the calculations of Subsection 2.4.1, Cases (1b), (1c), (4), (5). However, it is straightforward to check how our list of normal forms, as summarized in Examples 7.3 – 7.7, corresponds (modulo scalars) to the systems of canonical forms in [I], [E].

3. Topological considerations

We recall some well-known general facts on Grassmannian manifolds, and also recall from [L] some topological properties of real m-submanifolds of complex n-manifolds: the local property of “general position” for real submanifolds, and also global properties measured by characteristic classes. The Grassmannian constructions and formulas of [L], in the special case of the topology of real surfaces immersed in complex surfaces (m = n = 2), are also reviewed by [BF], [Bohr], [F], [ABKLR] §8.5.3. In this Section, we work with connected, m-dimensional, smooth manifolds M, not necessarily real analytic.

3.1. The Grassmannian construction.

For 0 < m < 2n, let G(m, ℝ^{2n}) denote the Grassmannian manifold of mdimensional real linear subspaces of ℝ^{2n}. The real dimension of G(m, ℝ^{2n}) is m · (2n − m).

Similarly, let SG(m, R^{2n}) denote the manifold of oriented m-dimensional subspaces of ℝ^{2n}. Its real dimension is also m·(2n−m). There is a two-to-one covering map F : SG(m, ℝ^{2n}) → G(m, ℝ^{2n}) given by forgetting the orientation, and an involution R : SG(m, ℝ^{2n}) → SG(m, ℝ^{2n}) given by reversing the orientation.

For any immersion ι of a real m-manifold M into ℝ^{2n}, we can define the Gauss map γ : M → G(m, ℝ^{2n}) : x ↦ T_{ι(x)}M. If M has an orientation, then the corresponding map is denoted γ_{s} : M → SG(m, ℝ^{2n}).

When m is even and there is a “complex structure operator,” J, a real linear map ℝ^{2n} → ℝ^{2n} such that J ∘ J = −𝟙, then G(m, ℝ^{2n}) contains C, the set of J-invariant subspaces. C is a submanifold: it is the image in G(m, ℝ^{2n}) of the inclusion embedding of the complex Grassmannian manifold ℂG(m/2, ℂ^{n}) of complex subspaces in ℂ^{n}.

In the case m = 2n − 2, the manifold ℂG(n − 1, ℂ^{n}) is homeomorphic to the complex projective space ℂP^{n−1}, and the real dimension of C is 2(n − 1), half the dimension 4(n − 1) of G(2n − 2, ℝ^{2n}). The inverse image F^{−1}(C) ⊆ SG(2n − 2, ℝ^{2n}) is a disjoint union C^{+} ∪ C^{−}, where C^{+} is the set of oriented subspaces whose orientation agrees with that given by the complex structure, and C^{−} is the set of subspaces where these orientations are opposite (so they could be called “anticomplex” subspaces). Each component C^{+} and C^{−} is the image of an embedding of ℂG(n − 1, ℂ^{n}), with real dimension, and codimension, equal to 2(n − 1). The C^{+} submanifold has a natural orientation.

For 0 < m < 2n and 0 ≤ j ≤ n, given an immersion of an m-submanifold ι : M → ℝ^{2n} = ℂ^{n}, let N_{j} denote the subset

N_{j} = {x ∈ M : dim_{ℝ}(T_{x}M ∩ J · T_{x}M) ≥ 2j},

so at a point x ∈ N_{j}, the tangent space T_{x}M contains a J-invariant subspace of real dimension 2j, which is a complex subspace of complex dimension j in ℂ^{n}.

For m = 2n − 2, N_{n−2} = M and N_{n−1} = γ^{−1}(C) is the CR singular locus. The immersion is in “general position” at x ∈ M if the Gauss map γ : M → G(2n − 2, ℝ^{2n}) meets C transversely at γ(x). M is trivially in general position at all its CR generic points, and the immersion is said to be in general position if it is in general position at every point. Then, by counting dimensions, an immersion in general position has only isolated CR singular points, and if M is compact, then the CR singular locus N_{n−1} is a finite set.

If M has an orientation, then the CR singular locus is the same set as (F ∘ γ_{s})^{−1}(C) = γ_{s}^{−1}(F^{−1}(C)) = γ_{s}^{−1}(C^{+} ∪ C^{−}), which is a disjoint union γ_{s}^{−1}(C^{+}) ∪ γ_{s}^{−1}(C^{−}). So, N_{n−1} = N_{n−1}^{+} ∪ N_{n−1}^{−}, where N_{n−1}^{+} = γ_{s}^{−1}(C^{+}) and N_{n−1}^{−} = γ_{s}^{−1}(C^{−}). The local and global notions of “general position” as defined previously are equivalent to the analogous transverse meeting of γ_{s} with C^{+} ∪ C^{−} in SG(2n − 2, ℝ^{2n}). At each point of N_{n−1}^{+} in general position, there is an oriented intersection number, ±1, of γ_{s} with the oriented submanifold C^{+}: the intersection number at γ_{s}(x) is be denoted the index, ind(x).

To define the intersection index at a point x of N_{n−1}^{−}, [L] makes a choice of orientation for C^{−} which is opposite to that induced by R : C^{+} → C^{−}. Then ind(x), the intersection number of γ_{s} with C^{−} at γ_{s}(x), is equal to the intersection number of R ∘ γ_{s} with C^{+} at R(γ_{s}(x)). Equivalently, if M^{−} denotes the manifold M with its orientation reversed, with Gauss map γ′_{s} : M^{−} → SG(2n − 2, ℝ^{2n}), then ind(x) is equal to the intersection number of γ′_{s} with C^{+} at γ′_{s}(x).

3.2. Bundle maps.

More generally, let M be a smooth, oriented manifold with real dimension 2n − 2, and let F be a smooth, oriented real vector bundle over M with 2n-dimensional fibers. Then the space of oriented real (2n − 2)-subspaces of fibers of F forms a “Grassmann bundle” SG(2n − 2, F) over M — on a local coordinate patch U of M where F can be trivialized as U × ℝ^{2n}, the Grassmann bundle is of the form U × SG(2n − 2, ℝ^{2n}). Suppose F admits a smooth complex structure operator J. Then for each point x ∈ M, the J_{x}- invariant subspaces form sets C_{x}^{+} and C_{x}^{−} in the fiber over x, giving a pair of smooth bundles C^{+}, C^{−} of complex Grassmannians over M; each total space has codimension 2n − 2 in the total space SG(2n − 2, F). If T is another smooth, oriented real vector bundle over M, with (2n − 2)-dimensional fibers, then a nonsingular bundle map μ : T → F induces a section γ_{μ} : M → SG(2n − 2, F) : x ↦ μ(T_{x}), generalizing the Gauss map. The transverse intersection of γ_{μ} with C^{+} and C^{−} defines a notion of “general position” for T, and the intersection numbers define the index of generally isolated points x where γ_{μ}(x) = μ(T_{x}) is a J_{x}-invariant subspace of F_{x}.

A special case of this bundle construction is where T is the tangent bundle T M of M, and ι is an immersion of M in an almost complex manifold A, with real dimension 2n and a smooth complex structure operator J on T A. Then the differential of ι defines a smooth bundle map μ : T M → F = ι^{∗}T A, inducing γ_{μ} : M → SG(2n − 2, F). A point x ∈ M is a CR singular point of ι if γ_{μ}(x) = μ(T_{x}M) is a J-invariant subspace of T_{ι(x)}A.

If, additionally, T A admits a positive definite Riemannian metric g, then there is an oriented real 2-plane normal bundle νM orthogonal to T M in ι^{∗}T A, and further, if g has the property that J is an isometry with respect to g, then T_{x}M is a complex hyperplane in ι^{∗}T_{x}A if and only if ν_{x}M is a complex line. Such a metric g can be chosen for any (T A, J), although we do not use it except to define the normal bundle.

3.3. Characteristic class formulas.

Following the notation of [F], for an immersion ι : M → A (or bundle map μ : T → F) as in the previous Subsection, we define index sums:

I_{+} = ∑_{x∈N_{n−1}^{+}} ind(x)

I_{−} = ∑_{x∈N_{n−1}^{−}} ind(x).

When M is compact and the immersion (or bundle map) is in general position, I_{+} and I_{−} are finite sums of ±1 terms. Then I_{+}, I_{−}, I_{+} +I_{−}, and I_{+} −I_{−} are all invariants of the homotopy class of ι (or μ). Reversing the orientation of M interchanges the values of I_{+} and I_{−}, so I_{+} + I_{−} is the same and I_{+} − I_{−} has the opposite sign.

If, instead of following [L] as in Subsection 3.1, we make the other choice of orientation for C^{−}, then I_{−} has the opposite sign and the quantities I_{+} + I_{−} and I_{+}−I_{−} are switched. Some of the following examples show that this is the choice of orientation that corresponds to index sums appearing in enumerative formulas of [Webster], [HL_{1}], [HL_{2}], [Domrin], and [C_{1}].

Our notation for characteristic classes in cohomology is copied from [L]: denote the Euler class of the tangent bundle T M by Ω, denote the Euler class of the normal bundle νM → M of the immersion by Ω̃, and denote the total Chern class of the pullback bundle ι^{∗}(T A, J) → M by 1 + c_{1} + c_{2} + … + c_{n}.

Proposition 3.1 ([L]). For an immersion ι of a compact, oriented (2n − 2)- manifold M in general position in an almost complex 2n-manifold A,

I_{+} = ∫_{M} \frac{1}{2} (Ω + ∑_{r=0}^{n−1} Ω̃^{r}c_{n−1−r}) .

Corollary 3.2. With the orientation convention for C^{−} as in [L],

I_{−} = ∫_{M} \frac{1}{2} (Ω + ∑_{r=0}^{n−1}(−1)^{r+1}Ω̃^{r}c_{n−1−r}) ,

I_{+} + I_{−} = ∫_{M} (Ω + ∑_{k=0}^{⌊n/2⌋−1} Ω̃^{2k+1}c_{n−2k−2}) ,

I_{+} − I_{−} = ∫_{M} ∑_{k=0}^{⌊(n−1)/2⌋} Ω̃^{2k}c_{n−2k−1}.

Proof. I_{−} is calculated by reversing the orientation of the tangent and normal bundles T M and νM, which switches the sign of Ω and Ω̃. Then, applying the formula from the Proposition, integrating over M^{−} gives the opposite of the integral over M.

Example 3.3. When A = ℂ^{n}, the Chern classes are trivial, so for M immersed in general position,

I_{+} = ∫_{M} \frac{1}{2} (Ω + Ω̃^{n−1}) ,
I_{−} = ∫_{M} \frac{1}{2} (Ω + (−1)^{n}Ω̃^{n−1}) .

If, in addition, ι is an embedding, then Ω̃ is the zero class, so I_{+} = I_{−} = \frac{1}{2}χ(M), where χ(M) is the Euler characteristic of M ([L] Theorem 4.10).

Example 3.4. For n = 2, ι is an immersion of a compact, oriented real surface M in a 4-manifold A with an almost complex structure J. Complex or anticomplex points with Lai’s index +1 are “elliptic,” or “hyperbolic” with index −1. Again following the notation of [F] and [ABKLR], if e_{+} (respectively, e_{−}) is the number of elliptic points with positively (negatively) oriented complex tangent spaces and h_{+} (h_{−}) is the number of positive (negative) hyperbolic points, and e = e_{+} + e_{−}, h = h_{+} + h_{−}, then for M in general position,

I_{+} = e_{+} − h_{+} = ∫_{M} \frac{1}{2} (Ω + c_{1} + Ω̃) ,
I_{−} = e_{−} − h_{−} = ∫_{M} \frac{1}{2} (Ω − c_{1} + Ω̃) ,
I_{+} + I_{−} = e − h = ∫_{M} (Ω + Ω̃) ,
I_{+} − I_{−} = (e_{+} − e_{−}) − (h_{+} − h_{−}) = ∫_{M} c_{1}.

The last formula was also proved by [Webster] and is a special case of a degeneracy locus formula of [HL_{2}] and [Domrin]. When ι is an embedding into A = ℂ^{2}, Ω̃ = 0, so the formulas are e_{+} − h_{+} = e_{−} − h_{−} = \frac{1}{2}χ(M), which were known to [Bishop] and [Wells] in the special cases where M is an embedded sphere or torus. Immersions with double points are considered by [BF] and [Bohr].

Example 3.5. For n = 3, ι is an immersion of a compact, oriented real 4manifold M in a 6-manifold A with an almost complex structure J. For M in general position, we adapt from the previous Example the notation e_{+}, h_{+} for counting elements of N_{2}^{+}, and e_{−}, h_{−} for elements of N_{2}^{−}. (However, the terms “elliptic” and “hyperbolic” are not adapted here from n = 2 to n ≥ 3, they appear in Section 7 in a more specific application.)

I_{+} = e_{+} − h_{+} = ∫_{M} \frac{1}{2} (Ω + c_{2} + Ω̃c_{1} + Ω̃^{2}) ,
I_{−} = e_{−} − h_{−} = ∫_{M} \frac{1}{2} (Ω − c_{2} + Ω̃c_{1} − Ω̃^{2}) ,
I_{+} + I_{−} = e − h = ∫_{M} (Ω + Ω̃c_{1}) ,
I_{+} − I_{−} = (e_{+} − e_{−}) − (h_{+} − h_{−}) = ∫_{M} (c_{2} + Ω̃^{2}) .

There are also formulas

(3.1)  I_{+} − I_{−} = ∫_{M} (c_{2} + p_{1}νM)
(3.2)  = ∫_{M} (c_{1}^{2} − c_{2} − p_{1}T M) ,

where p_{1} is the first Pontrjagin class of the normal bundle νM or tangent bundle T M. The equality (3.1) follows from the well-known identity of characteristic classes Ω̃^{2} = p_{1}νM. Formula (3.2) is a special case of the degeneracy locus formulas of [HL_{2}] and [Domrin] — the calculation establishing the equivalence of (3.1) and (3.2) appears in [C_{1}].

Example 3.6. Not every compact, oriented 4-manifold can be immersed in ℝ^{6} ≅ ℂ^{3} — for example, ℂP^{2} cannot ([Hirsch]). If such a 4-manifold M is immersed in general position in ℂ^{3}, then I_{+} + I_{−} = χ(M) and

I_{+} − I_{−} = ∫_{M} p_{1}νM = ∫_{M} −p_{1}T M = −p_{1}M,

the opposite of the first Pontrjagin number of M ([HL_{1}]). The number ∫_{M} p_{1}νM is also three times the algebraic number of triple points of an immersion in general position ([Herbert]). A compact, oriented 4-manifold admitting a CR generic immersion in ℂ^{3} must have χ(M) = p_{1}M = 0; the converse problem, finding a sufficient condition for the existence of a CR generic immersion, is considered by [JL].

Example 3.7. Consider the complex projective space A = ℂP^{3}, and a nonsingular, degree d complex hypersurface Y ⊆ ℂP^{3}. In terms of the hyperplane class H in the cohomology ring of ℂP^{3}, there are well-known formulas ([GH] §§1.1, 3.4, 4.6) for Chern classes: c_{2}(T ℂP^{3}|_{Y}) = 6H^{2}, c_{1}(T ℂP^{3}|_{Y}) = 4H, and c_{1}νY = dH. Since ∫_{Y} H^{2} = d,

∫_{Y} (c_{2}(T ℂP^{3}|_{Y}) + (c_{1}νY)^{2}) = 6d + d^{3},
∫_{Y} (c_{2}T Y + c_{1}νY · c_{1}(T ℂP^{3}|_{Y})) = χ(Y) + 4d^{2}.

Let M be a smooth real submanifold of ℂP^{3} isotopic to Y, that is, homotopic through a family of smooth embeddings. Then the Chern classes of Y pull back to classes on M, and if M is in general position, then

∫_{Y} (c_{2}(T ℂP^{3}|_{Y}) + (c_{1}νY)^{2}) = ∫_{M} (c_{2}(T ℂP^{3}|_{M}) + Ω̃^{2})
= 6d + d^{3} = I_{+} − I_{−},
∫_{Y} (c_{2}T Y + c_{1}νY · c_{1}(T ℂP^{3}|_{Y})) = ∫_{M} (Ω + Ω̃ · c_{1}(T ℂP^{3}|_{M}))
= χ(M) + 4d^{2} = I_{+} + I_{−}.

by the formulas from Example 3.5. These numbers are always positive; there is no CR generic submanifold of ℂP^{3} isotopic to a smooth complex hypersurface. One also expects that sufficiently nearby perturbations of Y would have only positively oriented complex tangents (N_{2}^{−} = Ø), so I_{−} = 0 and we recover the formula χ(Y) = χ(M) = d^{3} − 4d^{2} + 6d ([GH] p. 601).

In particular, consider the holomorphic embedding given in homogeneous coordinates by c : ℂP^{2} → ℂP^{3} : [z_{0} : z_{1} : z_{2}] → [z_{0} : z_{1} : z_{2} : 0], so the image Y is a complex hyperplane with degree d = 1. ℂP^{2} considered only as a smooth, oriented 4-manifold (forgetting its complex structure) has χ(ℂP^{2}) = 3. An embedding of M = ℂP^{2} as an oriented real submanifold in general position and isotopic to c has seven complex tangents, counted as an index sum with multiplicity according to either the I_{+} + I_{−} or I_{+} − I_{−} sign convention. In Section 6 we give a concrete example of such an isotopy from c to a smooth embedding ℂP^{2} → ℂP^{3} in general position with exactly seven CR singular points in N_{2}^{+}, each with index = +1: I_{+} = e_{+} = 7, h_{+} = I_{−} = e_{−} = h_{−} = 0.

Remark. If ι : ℂP^{2} → ℂP^{3} is any immersion (or any continuous map homotopic to an immersion), then ι is homotopic to either the embedding c from the above Example, or the composite c ∘ κ, where κ is the orientation-preserving involution [z_{0} : z_{1} : z_{2}] ↦ [z̄_{0} : z̄_{1} : z̄_{2}] ([Thomas], [LP]). Equivalently, either ι or ι ∘ κ is homotopic to c.

3.4. Local coordinates for the Grassmannian.

For 0 < m < 2n, each element v ∈ G(m, ℝ^{2n}) is the image of some linear map ℝ^{m} → ℝ^{2n} with standard matrix representation X_{2n×m} of rank m, and any two linear maps with the same image are right-equivalent (X ∼ XY for invertible Y_{m×m}). If v^{0} is the m-plane {(v_{1}, …, v_{m}, 0, …, 0)^{T} } in ℝ^{2n},

then it is the image of X^{0} = ( \begin{matrix}𝟙_{m×m} \\ 0 \end{matrix} )_{2n×m}, and any elements sufficiently

near v^{0} are the image of some linear map whose matrix representation can be column-reduced to the form

(3.3)  ( \begin{matrix}𝟙_{m×m} \\ V_{(2n−m)×m} \end{matrix} )_{2n×m} .

The matrices V in a neighborhood of the (2n − m) × m zero matrix form a local coordinate chart around v^{0} in G(m, ℝ^{2n}). The inverse image of a sufficiently small chart under F : SG(m, ℝ^{2n}) → G(m, ℝ^{2n}) gives a pair of charts in SG(m, ℝ^{2n}), one around each element of F^{−1}(v^{0}), the m-plane with its two possible orientations.

In the case where m is even and ℝ^{2n} has coordinates (x_{1}, y_{1}, …, x_{n}, y_{n})^{T} and a complex structure operator

J_{2n×2n} = ( \begin{matrix}0 & −1 \\ 1 & 0 \\ & & . & . & . \\ & & & & & 0 & −1 \\ & & & & & 1 & 0 \end{matrix} ) ,

consider the m-plane

v^{0} = {(x_{1}, y_{1}, …, x_{m/2}, y_{m/2}, 0, …, 0)^{T} }.

Since v^{0} is J-invariant, v^{0} ∈ C. Any element v of the intersection of C with the local coordinate chart around v^{0} would have matrix representation X =

( \begin{matrix}𝟙 \\ V \end{matrix} )_{2n×m}, such that J · X ∼ X. The equivalence

J · ( \begin{matrix}𝟙 \\ V \end{matrix} ) = ( \begin{matrix}J_{m×m} \\ J_{(2n−m)×(2n−m)} · V \end{matrix} )
∼ ( \begin{matrix}J \\ J · V \end{matrix} ) · (−J)_{m×m} = ( \begin{matrix}𝟙 \\ −J · V · J \end{matrix} )

shows v ∈ C ⟺ V = −J · V · J ⟺ V · J = J · V, that is, V is complex linear with respect to J_{m×m} and J_{(2n−m)×(2n−m)}.

For example, in the m = 4, n = 3 case, the coordinates near v^{0} in the

8-dimensional space G(4, ℝ^{6}) are of the form V = ( \begin{matrix}a_{1} & a_{2} & a_{3} & a_{4} \\ b_{1} & b_{2} & b_{3} & b_{4} \end{matrix} ), and

the elements of C have coordinates V satisfying

( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) · ( \begin{matrix}a_{1} & a_{2} & a_{3} & a_{4} \\ b_{1} & b_{2} & b_{3} & b_{4} \end{matrix} )

= ( \begin{matrix}a_{1} & a_{2} & a_{3} & a_{4} \\ b_{1} & b_{2} & b_{3} & b_{4} \end{matrix} ) · ( \begin{matrix}0 & −1 \\ 1 & 0 \\ & & 0 & −1 \\ & & 1 & 0 \end{matrix} ) .

This linear condition on a_{1}, …, b_{4} has a 4-dimensional solution subspace

{b_{1} = −a_{2}, b_{2} = a_{1}, b_{3} = −a_{4}, b_{4} = a_{3}}.

Now consider v^{0} with its orientation as a complex subspace of (ℝ^{6}, J), so v^{0} ∈ C^{+} ⊆ SG(4, ℝ^{6}). Also, instead of 2 × 4 matrices, we put the eight coordinate functions for SG(4, ℝ^{6}) near v^{0} in a column vector format, (a_{1}, a_{2}, a_{3}, a_{4}, b_{1}, b_{2}, b_{3}, b_{4})^{T}. Then, in this chart around v^{0}, C^{+} is the image of the linear map ℝ^{4} → ℝ^{8} with matrix representation

( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & −1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & −1 \\ 0 & 0 & 1 & 0 \end{matrix} ) = ( \begin{matrix}𝟙_{4×4} \\ J_{4×4} \end{matrix} ) .

Returning to the general situation of codimension 2 in ℂ^{n}, the above pattern

still holds, so that C^{+} is the column space of ( \begin{matrix}𝟙 \\ J \end{matrix} )_{2(2n−2)×(2n−2)} in the

coordinate chart around v^{0} in SG(2n − 2, ℝ^{2n}).

4. A reformulation of Garrity’s transversality criterion

Given a real (2n − 2)-submanifold of ℂ^{n} with a CR singular point, we now consider the problem of determining from the local defining equation whether M is in general position, as defined in Section 3, and if so, how the local defining equation determines the intersection index. The transversality problem was also considered by [Garrity] for real (2n − 2)-submanifolds of ℂ^{n}, using different methods but arriving at an equivalent result. Our result (Theorem 4.1) relates transversality to an expression in terms of the coefficient matrix notation from Section 2.

4.1. A determinantal formula.

We begin by assuming M is in standard position, given an orientation agreeing with the orientation of its complex tangent space at the origin. M can be described by a complex implicit equation (2.1) in a neighborhood Δ of 0:

(4.1) 0 = z_{n} − h(z, z̄) = z_{n} − (z^{T} Qz + z̄^{T} Rz + z̄^{T} Sz̄ + e(z, z̄)) .

If we now consider two real functions, f^{1}(x_{1}, y_{1}, …, x_{n−1}, y_{n−1}) = Re(h(z, z̄)) and f^{2}(x_{1}, …, y_{n−1}) = Im(h(z, z̄)), then the real (2n − 2)-manifold M has a local parametrization π with domain D ⊆ ℝ^{2n−2} and embedding target ℝ^{2n}:

(4.2)  π : (x_{1}, …, y_{n−1})^{T} ↦ (x_{1}, …, y_{n−1}, f^{1}, f^{2})^{T} .

The differential of this map assigns to each point z ∈ D a linear map from ℝ^{2n−2} to T_{π(z)}M ⊆ ℝ^{2n}; this linear map has matrix representation:

( \begin{matrix}& 𝟙_{(2n−2)×(2n−2)} \\ \frac{df^{1}}{dx_{1}} & ⋯ & \frac{df^{1}}{dy_{n−1}} \\ \frac{df^{2}}{dx_{1}} & ⋯ & \frac{df^{2}}{dy_{n−1}} \end{matrix} )_{(2n)×(2n−2)} .

This matrix is already in the form (3.3), so in the local coordinate systems π for M and V for SG(2n − 2, ℝ^{2n}), the oriented Gauss map has the form

γ_{s} : (x_{1}, y_{1}, …, x_{n−1}, y_{n−1})^{T} ↦ ( \begin{matrix}\frac{df^{1}}{dx_{1}} & ⋯ & \frac{df^{1}}{dy_{n−1}} \\ \frac{df^{2}}{dx_{1}} & ⋯ & \frac{df^{2}}{dy_{n−1}} \end{matrix} )_{2×(2n−2)} .

This map takes the CR singular point 0 ∈ M ⊆ (ℝ^{2n}, J) to the complex hyperplane v^{0} ∈ C^{+} ⊆ SG(2n − 2, ℝ^{2n}), with coordinates V = 0_{2×(2n−2)}. If we arrange the above two rows into column vector format (as in the end of Subsection 3.4), then the differential of the Gauss map at the origin has matrix representation:

( \begin{matrix}\frac{d}{dx_{1}} ( \frac{df^{1}}{dx_{1}} ) & \frac{d}{dy_{1}} ( \frac{df^{1}}{dx_{1}} ) & … & \frac{d}{dx_{n−1}} ( \frac{df^{1}}{dx_{1}} ) & \frac{d}{dy_{n−1}} ( \frac{df^{1}}{dx_{1}} ) \\ \frac{d}{dx_{1}} ( \frac{df^{1}}{dy_{1}} ) & \frac{d}{dy_{1}} ( \frac{df^{1}}{dy_{1}} ) & … & \frac{d}{dx_{n−1}} ( \frac{df^{1}}{dy_{1}} ) & \frac{d}{dy_{n−1}} ( \frac{df^{1}}{dy_{1}} ) \\ ⋮ & & & & ⋮ \\ \frac{d}{dx_{1}} ( \frac{df^{1}}{dx_{n−1}} ) & \frac{d}{dy_{1}} ( \frac{df^{1}}{dx_{n−1}} ) & … & \frac{d}{dx_{n−1}} ( \frac{df^{1}}{dx_{n−1}} ) & \frac{d}{dy_{n−1}} ( \frac{df^{1}}{dx_{n−1}} ) \\ \frac{d}{dx_{1}} ( \frac{df^{1}}{dy_{n−1}} ) & \frac{d}{dy_{1}} ( \frac{df^{1}}{dy_{n−1}} ) & … & \frac{d}{dx_{n−1}} ( \frac{df^{1}}{dy_{n−1}} ) & \frac{d}{dy_{n−1}} ( \frac{df^{1}}{dy_{n−1}} ) \\ \frac{d}{dx_{1}} ( \frac{df^{2}}{dx_{1}} ) & \frac{d}{dy_{1}} ( \frac{df^{2}}{dx_{1}} ) & … & \frac{d}{dx_{n−1}} ( \frac{df^{2}}{dx_{1}} ) & \frac{d}{dy_{n−1}} ( \frac{df^{2}}{dx_{1}} ) \\ \frac{d}{dx_{1}} ( \frac{df^{2}}{dy_{1}} ) & \frac{d}{dy_{1}} ( \frac{df^{2}}{dy_{1}} ) & … & \frac{d}{dx_{n−1}} ( \frac{df^{2}}{dy_{1}} ) & \frac{d}{dy_{n−1}} ( \frac{df^{2}}{dy_{1}} ) \\ ⋮ & & & & ⋮ \\ \frac{d}{dx_{1}} ( \frac{df^{2}}{dx_{n−1}} ) & \frac{d}{dy_{1}} ( \frac{df^{2}}{dx_{n−1}} ) & … & \frac{d}{dx_{n−1}} ( \frac{df^{2}}{dx_{n−1}} ) & \frac{d}{dy_{n−1}} ( \frac{df^{2}}{dx_{n−1}} ) \\ \frac{d}{dx_{1}} ( \frac{df^{2}}{dy_{n−1}} ) & \frac{d}{dy_{1}} ( \frac{df^{2}}{dy_{n−1}} ) & … & \frac{d}{dx_{n−1}} ( \frac{df^{2}}{dy_{n−1}} ) & \frac{d}{dy_{n−1}} ( \frac{df^{2}}{dy_{n−1}} ) \end{matrix} )|_{( 0 ⋮ 0 )}

= ( \begin{matrix}Hf^{1} \\ Hf^{2} \end{matrix} )_{2(2n−2)×(2n−2)} ,

where Hf^{1} and Hf^{2} are the real (2n − 2) × (2n − 2) real Hessian matrices of second derivatives, evaluated at (x_{1}, …, y_{n−1})^{T} = (0, …, 0)^{T}. The tangent space of the image γ_{s}(M) at γ_{s}(0) = v^{0} is spanned by the columns of this matrix, so in the coordinate chart around v^{0}, it is a (2n−2)-dimensional subspace that meets C^{+} transversely if the columns of this matrix are independent:

(4.3)  ( \begin{matrix}𝟙 & Hf^{1} \\ J & Hf^{2} \end{matrix} )_{2(2n−2)×2(2n−2)} .

So, M has a CR singularity in general position if and only if the determinant of the above matrix is nonzero. The intersection index at the origin is related to the sign of the determinant: ind = +1 for det > 0, ind = −1 for det < 0.

This product has the same determinant:

( \begin{matrix}𝟙 & 0 \\ 𝟙 & J \end{matrix} ) · ( \begin{matrix}𝟙 & Hf^{1} \\ J & Hf^{2} \end{matrix} ) = ( \begin{matrix}𝟙 & Hf^{1} \\ 0 & Hf^{1} + J · Hf^{2} \end{matrix} ) ,

so calculating the determinant reduces to a smaller, (2n − 2) × (2n − 2) real determinant, det(Hf^{1} + J · Hf^{2}).

In the m = 4, n = 3 case, the 4 × 4 matrix Hf^{1} + J · Hf^{2} is:

(4.4) ( \begin{matrix}f_{x_{1}x_{1}}^{1} − f_{y_{1}x_{1}}^{2} & f_{x_{1}y_{1}}^{1} − f_{y_{1}y_{1}}^{2} & f_{x_{1}x_{2}}^{1} − f_{y_{1}x_{2}}^{2} & f_{x_{1}y_{2}}^{1} − f_{y_{1}y_{2}}^{2} \\ f_{y_{1}x_{1}}^{1} + f_{x_{1}x_{1}}^{2} & f_{y_{1}y_{1}}^{1} + f_{x_{1}y_{1}}^{2} & f_{y_{1}x_{2}}^{1} + f_{x_{1}x_{2}}^{2} & f_{y_{1}y_{2}}^{1} + f_{x_{1}y_{2}}^{2} \\ f_{x_{2}x_{1}}^{1} − f_{y_{2}x_{1}}^{2} & f_{x_{2}y_{1}}^{1} − f_{y_{2}y_{1}}^{2} & f_{x_{2}x_{2}}^{1} − f_{y_{2}x_{2}}^{2} & f_{x_{2}y_{2}}^{1} − f_{y_{2}y_{2}}^{2} \\ f_{y_{2}x_{1}}^{1} + f_{x_{2}x_{1}}^{2} & f_{y_{2}y_{1}}^{1} + f_{x_{2}y_{1}}^{2} & f_{y_{2}x_{2}}^{1} + f_{x_{2}x_{2}}^{2} & f_{y_{2}y_{2}}^{1} + f_{x_{2}y_{2}}^{2} \end{matrix} ) ,

evaluated at (x_{1}, y_{1}, x_{2}, y_{2})^{T} = (0, 0, 0, 0)^{T}. These real number entries can be expressed in terms of the derivatives at the origin of the original function h(z, z̄):

2( \begin{matrix}Re(h_{z_{1}z̄_{1}} +h_{z̄_{1}z̄_{1}}) & Im(−h_{z_{1}z̄_{1}} +h_{z̄_{1}z̄_{1}}) & Re(h_{z̄_{1}z_{2}} +h_{z̄_{1}z̄_{2}}) & Im(−h_{z̄_{1}z_{2}} +h_{z̄_{1}z̄_{2}}) \\ Im(h_{z_{1}z̄_{1}} +h_{z̄_{1}z̄_{1}}) & Re(h_{z_{1}z̄_{1}} − h_{z̄_{1}z̄_{1}}) & Im(h_{z̄_{1}z_{2}} +h_{z̄_{1}z̄_{2}}) & Re(h_{z̄_{1}z_{2}} − h_{z̄_{1}z̄_{2}}) \\ Re(h_{z_{1}z̄_{2}} +h_{z̄_{1}z̄_{2}}) & Im(−h_{z_{1}z̄_{2}} +h_{z̄_{1}z̄_{2}}) & Re(h_{z̄_{2}z_{2}} +h_{z̄_{2}z̄_{2}}) & Im(−h_{z̄_{2}z_{2}} +h_{z̄_{2}z̄_{2}}) \\ Im(h_{z_{1}z̄_{2}} +h_{z̄_{1}z̄_{2}}) & Re(h_{z_{1}z̄_{2}} − h_{z̄_{1}z̄_{2}}) & Im(h_{z̄_{2}z_{2}} +h_{z̄_{2}z̄_{2}}) & Re(h_{z̄_{2}z_{2}} − h_{z̄_{2}z̄_{2}}) \end{matrix}),

evaluated at z = (0, 0)^{T}. The (3, 1) entry can be checked, for example, by the following calculation which is similar to the derivation of all the other entries:

2 Re(h_{z_{1}z̄_{2}} + h_{z̄_{1}z̄_{2}})

= 2 Re (\frac{1}{2}(\frac{∂}{∂x_{2}} + i\frac{∂}{∂y_{2}} )\frac{1}{2}(\frac{∂}{∂x_{1}} − i\frac{∂}{∂y_{1}} )(f^{1} + if^{2})

+\frac{1}{2}(\frac{∂}{∂x_{2}} + i\frac{∂}{∂y_{2}} )\frac{1}{2}(\frac{∂}{∂x_{1}} + i\frac{∂}{∂y_{1}} )(f^{1} + if^{2}))

= \frac{1}{2}(f_{x_{1}x_{2}}^{1} + f_{y_{1}y_{2}}^{1} − f_{x_{1}y_{2}}^{2} + f_{y_{1}x_{2}}^{2} + f_{x_{1}x_{2}}^{1} − f_{y_{1}y_{2}}^{1} − f_{x_{1}y_{2}}^{2} − f_{y_{1}x_{2}}^{2})
= f_{x_{1}x_{2}}^{1} − f_{x_{1}y_{2}}^{2}.

Note that the entries in the above matrix do not depend on the second zderivatives h_{z_{j}z_{k}}, which are determined by the coefficient matrix Q in (4.1). This agrees with the notion that transversality and the index should not depend on the local holomorphic coordinate system, since it was shown in Section 2 how the coefficients Q could be arbitrarily altered by holomorphic coordinate changes. However, it is not as easy to see from the form of (4.3) or (4.4) that they do not depend on the Q coefficients. We also see in the above matrix that the entries depend only on the coefficients from R and S in the expression (4.1) for h(z, z̄), and not on e(z, z̄) = O(3).

By Lemma 4.3 (the proof of which is left to Subsection 4.3), the determinant of the above matrix is equal to

2^{4} det ( \begin{matrix}R & 2S \\ \overline{2S} & \overline{R} \end{matrix} ) .

The above calculations (including the Lemma) generalize to other dimensions n, and the index formula is even simpler with the defining equation in the form (2.12), with (n − 1) × (n − 1) complex symmetric coefficient matrix P = 2S̄:

Theorem 4.1. Given a real (2n−2)-submanifold M in ℂ^{n} with a CR singular point in standard position and local defining equation:

(4.5)  z_{n} = z̄^{T} Rz + Re (z^{T} P z) + e(z, z̄),

then M is in general position if and only if the matrix

Γ = ( \begin{matrix}R & \overline{P} \\ P & \overline{R} \end{matrix} )

is nonsingular. Further, if M is given an orientation agreeing with the orientation of the complex (z_{1}, …, z_{n−1})-hyperplane tangent to M at 0, then the intersection index (±1) is the sign of the determinant det(Γ).

Example 4.2. In the n = 2 case, if M is in standard position, oriented to agree with the orientation of the z_{1}-axis near the origin, and has defining equation

z_{2} = z_{1}z̄_{1} + γ_{1}z_{1}^{2} + γ̄_{1}z̄_{1}^{2} + e(z_{1}, z̄_{1}) = z_{1}z̄_{1} + Re(2γ_{1}z_{1}^{2}) + e(z_{1}, z̄_{1}), γ_{1} ∈ ℂ,

then the index is the sign of det ( \begin{matrix}1 & 2γ̄_{1} \\ 2γ_{1} & 1 \end{matrix} ) = 1 − 4γ_{1}γ̄_{1}, which is +1 for

0 ≤ |γ_{1}| < \frac{1}{2} and −1 for |γ_{1}| > \frac{1}{2}. M is not in general position for |γ_{1}| = \frac{1}{2} (the “parabolic” case).

For n in general, the real number value of det(Γ) is not an invariant under holomorphic transformations, only its sign is invariant. Generalizing the action of the group from (2.13) to n dimensions,

(4.6)  (R, P) ↦ (cĀ^{T} RA, c̄A^{T} P A),

for A = A_{(n−1)×(n−1)}, c = c_{n,n}. The matrix Γ and its determinant transform as:

Γ ↦ ( \begin{matrix}cĀ^{T} RA & \overline{c̄A^{T} P A} \\ c̄A^{T} P A & \overline{cĀ^{T} RA} \end{matrix} )

(4.7)  = ( \begin{matrix}c𝟙 & 0 \\ 0 & c̄𝟙 \end{matrix} ) ( \begin{matrix}Ā^{T} & 0 \\ 0 & A^{T} \end{matrix} ) ( \begin{matrix}R & \overline{P} \\ P & \overline{R} \end{matrix} ) ( \begin{matrix}A & 0 \\ 0 & Ā \end{matrix} ) ,
det(Γ) ↦ |c|^{2(n−1)}| det A|^{4} det ( \begin{matrix}R & \overline{P} \\ P & \overline{R} \end{matrix} ) .

Even if the determinant is zero, (4.7) shows that the rank of Γ is a biholomorphic invariant.

It also follows from the transformation formula that the vector

(| det(R)|^{2}, | det(P)|^{2}, det(Γ)) ∈ ℝ^{3}

has an invariant direction (the expression is invariant modulo positive scalar

multiplication), and ratios such as \frac{det(Γ)}{| det(R)|^{2}} are numerical invariants (when

well-defined).

4.2. Invariants at flat points.

In the holomorphically flat case (as in Subsections 2.2, 2.5), where in some local coordinates h(z, z̄) is real valued and so M is a real (2n−2)-hypersurface inside ℝ^{2n−1} = {y_{n} = 0}, and f^{2} = 0 in (4.2), the (2n − 2) × (2n − 2) matrix Hf^{1}+J ·Hf^{2} (4.4) is just the Hessian Hf^{1}. Since Hf^{1} is a real symmetric matrix, it has all real eigenvalues, and it follows from the above construction and the Proof of Lemma 4.3 that Γ is Hermitian symmetric, with real eigenvalues equal to \frac{1}{2} times the eigenvalues of Hf^{1}. When M is oriented as in Theorem 4.1, so the normal vector at the origin is in the positive x_{n} direction, the Hessian is exactly the matrix representation of the Weingarten shape operator at the origin ([Thorpe]). Its determinant is the Gauss-Kronecker curvature (the product of the 2n − 2 real eigenvalues, which are the principal curvatures). So, when M is a hypersurface in ℝ^{2n−1}, in general position, and positively oriented at a CR singular point, the index and the curvature have the same sign, which is [L] Lemma 4.11. If we consider holomorphic coordinate changes that preserve the property that M is contained in ℝ^{2n−1} (as in Subsection 2.5), then the matrix A in the transformations (4.6), (4.7) may be arbitrary, but if R ≠ 0, then c must be real. The A part acts as a Hermitian congruence transformation on Γ, preserving its rank ρ(Γ) and signature (p, q), but if c is negative, then the signature is switched to (q, p). For R = 0, the transformation (4.7) is a Hermitian congruence transformation of Γ for any complex c ≠ 0: instead of using the matrix factorization as in (4.7), let c = ρe^{iθ}, ρ > 0, 0 ≤ θ < 2π, in order to choose a square root, and then use the matrix

( \begin{matrix}\sqrt{ρ}e^{−iθ/2}A & 0 \\ 0 & \sqrt{ρ}e^{iθ/2}Ā \end{matrix} ) on the right.

We can conclude that for M ⊆ ℝ^{2n−1}, the rank ρ(Γ) and the quantity σ(Γ) = |p − q| (from which one can recover the signature, modulo switching, of Γ) are invariant under holomorphic transformations preserving the real valued property of h(z, z̄). Since the action of (c, A) on Γ is not affected by the higher degree terms, ρ(Γ) and σ(Γ) are also invariants of a defining function h(z, z̄) in a quadratically flat normal form, under holomorphic transformations preserving the property of being in a quadratically flat normal form.

For Hermitian R, if we think of the quadratic part of (4.5),

z̄^{T} Rz + Re (z^{T} P z) ,

as a real valued quadratic form on ℝ^{2n−2}, then its zero locus is a real algebraic variety. The dimension of this variety, and whether it is reducible or irreducible, are properties that are invariant under scalar multiplication of the form, and also under real linear coordinate changes of the domain ℝ^{2n−2}. In particular, they are also invariants under complex linear transformations z ↦ Az of ℂ^{n−1}. Real valued quadratic forms on ℂ^{n−1} are considered by [CS], where the zero set is called a “quadratic cone” if it is irreducible and has dimension 2n − 3. The action of scalar multiplication and complex linear transformations A on the set of equations of quadratic cones in ℂ^{n−1} is the same as the above action (4.6) on the set of matrix pairs (R, P) appearing in a quadratically flat normal form.

In the n = 3 case, a list of equivalence classes of real valued quadratic forms defining quadratic cones in ℂ^{2} is given by [CS], and each type of cone corresponds to one of the normal forms for 2 × 2 matrix pairs (R, P) computed in Subsection 2.4.1. Our list of normal forms for pairs (R, P), with R Hermitian, (summarized in Examples 7.3 – 7.7) is longer since [CS] excludes the reducible and lower-dimensional varieties.

4.3. Some matrix calculations.

Let R = ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) and S = ( \begin{matrix}a & b \\ c & d \end{matrix} ) be 2 × 2 matrices with arbitrary

complex entries. Consider the following 4 × 4 matrices with real entries:

R′ = ( \begin{matrix}Re(α) & − Im(α) & Re(β) & − Im(β) \\ Im(α) & Re(α) & Im(β) & Re(β) \\ Re(γ) & − Im(γ) & Re(δ) & − Im(δ) \\ Im(γ) & Re(γ) & Im(δ) & Re(δ) \end{matrix} )

S′ = ( \begin{matrix}Re(a) & Im(a) & Re(b) & Im(b) \\ Im(a) & − Re(a) & Im(b) & − Re(b) \\ Re(c) & Im(c) & Re(d) & Im(d) \\ Im(c) & − Re(c) & Im(d) & − Re(d) \end{matrix} ) .

Lemma 4.3. det(R′ + S′)_{4×4} = det ( \begin{matrix}R & S \\ \overline{S} & \overline{R} \end{matrix} )_{4×4}.

Proof. Let

K = \frac{\sqrt{2}}{2} ( \begin{matrix}1 & i & 0 & 0 \\ 0 & 0 & 1 & i \\ 1 & −i & 0 & 0 \\ 0 & 0 & 1 & −i \end{matrix} ) .

Then K is a unitary matrix with det(K) = 1. A calculation shows that:

KR′K̄^{T} = ( \begin{matrix}R & 0 \\ 0 & \overline{R} \end{matrix} ) ,
KS′K̄^{T} = ( \begin{matrix}0 & S \\ \overline{S} & 0 \end{matrix} ) .

Since det(R′ + S′) = det(K(R′ + S′)K̄^{T}), the claimed formula follows.

The Lemma generalizes to complex n × n matrices and the corresponding 2n × 2n matrices following the same pattern. In the n = 1 case, the analogous

matrix K is K = \frac{\sqrt{2}}{2} ( \begin{matrix}1 & i \\ 1 & −i \end{matrix} ), for n = 3,

K = \frac{\sqrt{2}}{2} ( \begin{matrix}1 & i & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & i & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & i \\ 1 & −i & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & −i & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & −i \end{matrix} ) ,

etc. Even without the result of the Lemma, it is easy to see that elementary

identities imply ( \begin{matrix}R & S \\ \overline{S} & \overline{R} \end{matrix} )_{2n×2n} has a real determinant. In the application

of the Lemma in Subsection 4.1, the S block is assumed to be symmetric — however, the Lemma does not need that assumption.

5. Complexification

We return to the description of M ⊆ ℝ^{2n} = ℂ^{n} as the image of a real analytic parametric map, with domain D ⊂ ℝ^{2n−2} and target ℂ^{n}. Re-writing Equation (4.2) in complex form gives:

z ↦ (z, h(z, z̄)) = (z_{1}, …, z_{n−1}, h(z_{1}, z̄_{1}, …, z_{n−1}, z̄_{n−1})).

The following complex analytic map, with domain D_{c} ⊆ ℂ^{2n−2} (as in (2.3)) and target ℂ^{n}, is a complexification of the above parametric map:

(z_{1}, …, z_{n−1}, w_{1}, …, w_{n−1}) ↦ (z_{1}, …, z_{n−1}, h(z_{1}, w_{1}, …, z_{n−1}, w_{n−1})).

The coordinates w = (w_{1}, …, w_{n−1}) are new complex variables; the new map restricted to w = z̄ (by substitution in the series expansion of h(z, z̄), as in (2.2)) is exactly the original map. A geometric interpretation of such a complexification construction (as the composite of a holomorphic embedding D_{c} → ℂ^{2n} and a linear projection ℂ^{2n} → ℂ^{n}) is given in [C_{3}] §4; the n = 2 case is used in [MW]. The complex map is singular at the origin (its complex Jacobian drops rank there since h(z, w) has no linear terms).

The origin-preserving local biholomorphic transformations of ℂ^{n}, z̃ = Cz⃗+ p⃗(z⃗), as in (2.9), act on the function h(z, z̄); this induces an action on the complex map (z, w) ↦ (z, h(z, w)). This group of transformations is a subgroup of a larger transformation group, which acts on the set of (germs at the origin of) maps ℂ^{2n−2} → ℂ^{n}, by composition with origin-preserving biholomorphic transformations of both the domain and the target. The algebraic interpretation of the complexification construction is that the larger group allows the transformation of the z and w (formerly z̄) variables independently. Any invariants under the larger group action are also invariants under the action of the subgroup.

In the case n = 2, the map (z_{1}, w_{1}) ↦ (z_{1}, h(z_{1}, w_{1})) can be put into one of three normal forms under the larger group: (z_{1}, w_{1}^{2} +O(3)), (z_{1}, z_{1}w_{1} +O(3)), (z_{1}, O(3)). As described in [C_{3}], the first two cases correspond, respectively, to Whitney’s fold and cusp singularities of maps ℂ^{2} → ℂ^{2}. In the fold case, a point near the origin in the target ℂ^{2} has two inverse image points.

In the case n = 3, the map D_{c} → ℂ^{3}:

(z_{1}, z_{2}, w_{1}, w_{2}) ↦ (z_{1}, z_{2}, h(z_{1}, w_{1}, z_{2}, w_{2}))

can be transformed to (z̃_{1}, z̃_{2}, q̃(z̃, w̃) + O(3)), where q̃(z̃, w̃) is the quadratic part, falling into one of the following six normal forms (the calculation is omitted):

w̃_{1}^{2} + w̃_{2}^{2}, z̃_{2}w̃_{2} + w̃_{1}^{2}, w̃_{1}^{2}, z̃_{1}w̃_{1} + z̃_{2}w̃_{2}, z̃_{1}w̃_{1}, 0.

With h(z, z̄) of the form (2.1), the rank of the coefficient matrix S is an invariant of the complexification under the large group, and so is the rank of (R|S)_{2×4}. These two numbers uniquely determine the equivalence class of the quadratic part under the larger group; the rank of R is not an invariant. The first of the above six cases is the generic one, where ρ(S) = 2 and the inverse image of a point near the origin in the target ℂ^{3} is a complex analytic curve in D_{c} ⊆ ℂ^{4}.

6. Various global examples

Here we collect some examples of compact real 4-manifolds embedded in complex 3-manifolds.

Example 6.1. The 4-sphere S^{4} has a real algebraic embedding in a real hyperplane ℝ^{5} ⊆ ℂ^{3}, which is (globally) holomorphically flat. For positive coefficients d_{1}, d_{2}, d_{3}, d_{4}, d_{5}, the implicit equation

d_{1}x_{1}^{2} + d_{2}y_{1}^{2} + d_{3}x_{2}^{2} + d_{4}y_{2}^{2} + d_{5}x_{3}^{2} = 1

defines an ellipsoidal hypersurface in the real hyperplane {y_{3} = 0}. There are exactly two CR singularities, where the tangent space is parallel to the (z_{1}, z_{2})- subspace. The two points are holomorphically equivalent to each other, and the two local real defining equations can be put into a complex normal form (2.12) with N = 𝟙, and P real diagonal. The Hessian at each point is definite, with the entries of P in the interval [0, 1), depending on d_{1}, …, d_{5}. Putting an orientation on S^{4} induces opposite orientations at the two points (one complex, the other anticomplex), so I_{+} = I_{−} = 1, consistent with the characteristic class formulas from Subsection 3.3: for any immersion of S^{4} in general position in ℂ^{3}, I_{+} + I_{−} = χ(S^{4}) = 2, and I_{+} − I_{−} = −p_{1}S^{4} = 0. For n ≥ 3, spheres S^{2n−2} in ℂ^{n} are considered by [DTZ_{2}].

Example 6.2. Every compact, oriented, three-dimensional, smooth manifold M^{3} admits a smooth immersion in ℝ^{4}, τ_{1} : M^{3} → ℝ^{4} ([Hirsch], [JL]). Not every such 3-manifold M^{3} admits an embedding, but the manifolds S^{3}, S^{2} × S^{1}, and S^{1} × S^{1} × S^{1} all can be embedded as hypersurfaces of revolution in ℝ^{4}. There is an immersion (but not an embedding) of ℝP^{3} in ℝ^{4} ([Hirsch], [M]). Also consider any oriented immersion of the circle, τ_{2} : S^{1} → ℝ^{2}. For any (almost) complex structures on ℝ^{4} and ℝ^{2}, τ_{1} is a CR regular immersion, τ_{2} is a totally real immersion, and the product τ_{1} × τ_{2} : M^{3} × S^{1} → ℝ^{4} × ℝ^{2} is an oriented, CR regular immersion (with respect to the product complex structure). The index sums I_{+} = I_{−} = 0 are consistent with the topological formulas from Subsection 3.3, since χ(M^{3} × S^{1}) and p_{1}(M^{3} × S^{1}) are both zero.

Example 6.3. The 4-manifold S^{2} × S^{2} does not admit any CR regular immersion in ℂ^{3}; an immersion in general position has I_{+} +I_{−} = χ(S^{2} ×S^{2}) = 4 and I_{+}−I_{−} = −p_{1}(S^{2}×S^{2}) = 0, so I_{+} = I_{−} = 2. We consider a real algebraic embedding in ℝ^{3} × ℝ^{3}, given by a product of ellipsoids:

(6.1)  {ax_{1}^{2} + by_{1}^{2} + cx_{3}^{2} = 1,
dx_{2}^{2} + ey_{2}^{2} + fy_{3}^{2} = 1} ,

with positive coefficients a, …, f. There are exactly four CR singularities, at the points with z_{1} = z_{2} = 0. Solving the real defining equations for x_{3} and y_{3}, setting z_{3} = x_{3} + iy_{3}, and translating the CR singularities into standard position, the local equation at each point is real analytic:

x_{3} + iy_{3} = \frac{η_{1}}{\sqrt{c}} ((−1) + \sqrt{1 − ax_{1}^{2} − by_{1}^{2}})

+i\frac{η_{2}}{\sqrt{f}} ((−1) + \sqrt{1 − dx_{2}^{2} − ey_{2}^{2}}) ,

z_{3} = \frac{η_{1}}{2\sqrt{c}} (a · (\frac{z_{1} + z̄_{1}}{2} )^{2} + b · (\frac{z_{1} − z̄_{1}}{2i} )^{2})

+i\frac{η_{2}}{2\sqrt{f}} (d · (\frac{z_{2} + z̄_{2}}{2} )^{2} + e · (\frac{z_{2} − z̄_{2}}{2i} )^{2}) + O(4),

where the four CR singular points correspond to the four sign choices η_{1} = ±1, η_{2} = ±1. After a transformation (2.5), the equation can be written in the form (2.12), with diagonal coefficient matrices (R, P):

z_{3} = (z̄_{1}, z̄_{2}) ( \begin{matrix}η_{1} \frac{a+b}{4\sqrt{c}} & 0 \\ 0 & iη_{2} \frac{d+e}{4\sqrt{f}} \end{matrix} ) ( \begin{matrix}z_{1} \\ z_{2} \end{matrix} )

+ Re ((z_{1}, z_{2}) ( \begin{matrix}η_{1} \frac{a−b}{4\sqrt{c}} & 0 \\ 0 & η_{2}i\frac{d−e}{4\sqrt{f}} \end{matrix} ) ( \begin{matrix}z_{1} \\ z_{2} \end{matrix} )) + O(4).

The normal form for the coefficient matrix pair, from Case (1a) of Subsection 2.4.1, has the same form for all four points: for η_{1} = η_{2},

(R, P) ∼ (( \begin{matrix}1 & 0 \\ 0 & i \end{matrix} ) , ( \begin{matrix}\frac{|a−b|}{a+b} & 0 \\ 0 & \frac{|d−e|}{d+e} \end{matrix} )) ,

and for η_{1} = −η_{2},

(R, P) ∼ (( \begin{matrix}1 & 0 \\ 0 & i \end{matrix} ) , ( \begin{matrix}\frac{|d−e|}{d+e} & 0 \\ 0 & \frac{|a−b|}{a+b} \end{matrix} )) ,

not depending on c, f. The nonnegative diagonal entries are local biholomorphic invariants. For any values of a, b, d, e, η_{1}, η_{2}, the matrix pair satisfies det(Γ) > 0, so the index is +1 for each CR singular point; there are two complex points and two anticomplex points.

We also note that this particular embedding of S^{2} × S^{2} is contained in the (Levi nondegenerate) real hypersurface

{ax_{1}^{2} + by_{1}^{2} + cx_{3}^{2} + dx_{2}^{2} + ey_{2}^{2} + fy_{3}^{2} = 2} ,

an ellipsoid in ℂ^{3}. So, the 4-manifold S^{2} × S^{2} admits some topological embedding as a hypersurface in ℝ^{5}, but the extrinsic geometry of this product embedding (6.1) is that globally, it is not contained in a Levi flat hypersurface, and locally, it is not quadratically flat at its CR singular points.

Example 6.4. Consider M = ℂP^{2} with homogeneous coordinates [z_{0} : z_{1} : z_{2}] and A = ℂP^{3} with coordinates [Z_{0} : Z_{1} : Z_{2} : Z_{3}]. For each t ∈ ℝ, let

ι_{t} : ℂP^{2} → ℂP^{3} :
(6.2)  [z_{0} : z_{1} : z_{2}] ↦ [z_{0} · P : z_{1} · P : z_{2} · P : t · Q],

where P is the polynomial expression

P = P(z_{0}, z_{1}, z_{2}) = 6z_{0}z̄_{0} + z_{1}z̄_{1} + 6z_{2}z̄_{2}

and Q is the polynomial expression

Q = Q(z_{0}, z_{1}, z_{2}) = 2z_{0}^{2}z̄_{1} + 2z_{0}z_{1}z̄_{2} − z_{0}z_{2}z̄_{1} + 2z_{1}z_{2}z̄_{0}.

Note that for any [z_{0} : z_{1} : z_{2}], P ≠ 0, and the first three components in the RHS of (6.2) have no common zeros. The formula (6.2) also has a homogeneity property:

ι_{t}([λ · z_{0} : λ · z_{1} : λ · z_{2}]) = [λ^{2}λ̄ · z_{0} · P : λ^{2}λ̄ · z_{1} · P : λ^{2}λ̄ · z_{2} · P : λ^{2}λ̄ · t · Q],

so ι_{t} is well-defined. Observe that for t = 0,

ι_{0}([z_{0} : z_{1} : z_{2}]) = [z_{0} · P : z_{1} · P : z_{2} · P : 0 · Q] = [z_{0} : z_{1} : z_{2} : 0]

is exactly the embedding c from Example 3.7. To show that for each t, ι_{t} is a real analytic embedding, and that the family ι_{t} is real analytic in t (so that this construction is an isotopy as in Example 3.7, and an “unfolding” as in [C_{3}]), we view ι_{t} in local affine coordinate charts.

The restriction of ι_{t} to the {z_{0} = 1} neighborhood has image contained in the {Z_{0} ≠ 0} neighborhood of the target ℂP^{3}, and is given by the formula:

[1 : z_{1} : z_{2}] ↦ [1 : z_{1} : z_{2} : \frac{t · Q(1, z_{1}, z_{2})}{P(1, z_{1}, z_{2})}] ,
(6.3)  (z_{1}, z_{2}) ↦ (z_{1}, z_{2}, \frac{t · (2z̄_{1} + 2z_{1}z̄_{2} − z_{2}z̄_{1} + 2z_{1}z_{2})}{6 + z_{1}z̄_{1} + 6z_{2}z̄_{2}}) .

This is a graph over the (z_{1}, z_{2})-hyperplane of a rational (in z, z̄) function F_{t}(z_{1}, z̄_{1}, z_{2}, z̄_{2}) = t·Q/P with a nonvanishing denominator, so this restriction of ι_{t} is a real analytic embedding. The dependence on t is real analytic, where ι_{0} is just the graph of the constant function 0. For each t ≠ 0, the graph is a smooth real submanifold M_{t} in ℂ^{3}, which is not a complex submanifold but which inherits its orientation from the (z_{1}, z_{2})-hyperplane. Its CR singularities can be located by solving the system of two complex equations

(6.4)  \frac{d}{dz̄_{1}} F_{t} = 0, \frac{d}{dz̄_{2}} F_{t} = 0.

The solution set does not depend on t (for t ≠ 0), and it is easy to check that each of the following five points (ζ_{1}, ζ_{2}) in the domain is a zero of the z̄-derivatives:

\{(0, 2), (\sqrt{3}, 1), (−\sqrt{3}, 1), (3i, −1), (−3i, −1)\}.

The points (ζ_{1}, ζ_{2}, F_{t}(ζ_{1}, ζ̄_{1}, ζ_{2}, ζ̄_{2})) are the CR singularities of M_{t}, each with a positively oriented complex tangent space. Given F_{t}, calculations with the assistance of [Maple] found the above five points by solving four real equations in four real unknowns, and (omitting the details) verified that these five points are the only solutions of (6.4) in this neighborhood.

The challenge in constructing this example by making a good choice for the above P and Q is to find coefficients which are simple and sparse enough so that solving the system is a tractable computation with a numerically exact solution set, but not so simple that M_{t} is not in general position and has a CR singularity with a degenerate normal form (ind ≠ ±1).

To calculate the index of the CR singular point at (ζ_{1}, ζ_{2}, F_{t}), we use Theorem 4.1 and find the sign of det(Γ) — this is where we need the exact coordinates of the CR singularities. By the form of (6.3), it is enough to check the t = 1 case, since for t ≠ 0, M_{t} is related to M_{1} by an invertible complex linear transformation of ℂ^{3}.

Corresponding to the solution (0, 2), translating the CR singular point to the origin in ℂ^{3} gives a multivariable Taylor expansion:

F_{1}(z_{1}, z̄_{1}, z_{2} + 2, \overline{z_{2} + 2}) = \frac{4}{15}z_{1} − \frac{1}{25}z_{1}z_{2} − \frac{1}{25}z_{1}z̄_{2} − \frac{1}{30}z_{2}z̄_{1} + O(3).

The linear term can be eliminated by a complex linear transformation z̃_{3} = z_{3} + c_{31}z_{1}, which does not change the quadratic or higher degree terms and brings M_{1} into standard position (2.1). The (R, S) coefficient matrix pair sat-

isfies S = 0_{2×2} and det(R) ≠ 0, so Γ = ( \begin{matrix}R & 0 \\ 0 & R̄ \end{matrix} ) and det(Γ) = | det(R)|^{2} >

0. This CR singularity has index +1.

Similarly, corresponding to the solution (\sqrt{3}, 1), translating to the origin gives the series expansion:

F_{1}(z_{1} + \sqrt{3}, \overline{z_{1} + \sqrt{3}}, z_{2} + 1, \overline{z_{2} + 1}) − \frac{\sqrt{3}}{3}

= \frac{1}{5}z_{1} − \frac{\sqrt{3}}{15}z_{2} − \frac{\sqrt{3}}{75}z_{1}^{2} + \frac{1}{15}z_{1}z_{2} + \frac{2\sqrt{3}}{75}z_{2}^{2}

−\frac{8\sqrt{3}}{225}z_{1}z̄_{1} + \frac{4}{75}z_{1}z̄_{2} − \frac{4}{75}z_{2}z̄_{1} − \frac{8\sqrt{3}}{75}z_{2}z̄_{2} + O(3).

Again in this case, the holomorphic terms are irrelevant, S = 0, and det(R) ≠ 0, so the CR singularity has index +1. Each of the remaining three points, by a similar (but omitted) calculation, also has index +1, so M_{1} is in general position.

It remains to check the points “at infinity,” where z_{0} = 0 and ι_{t} restricts to a holomorphic linear embedding

[0 : z_{1} : z_{2}] ↦ [0 : z_{1} · P : z_{2} · P : t · 0] = [0 : z_{1} : z_{2} : 0].

This restriction is one-to-one and misses the image of F_{t} in the {Z_{0} = 1} affine neighborhood, which shows that ι_{t} is one-to-one for each t.

To look for more CR singularities on the line at infinity, we consider the restriction of ι_{t} to another affine coordinate chart. The restriction of ι_{t} to the {z_{1} = 1} neighborhood has image contained in the {Z_{1} ≠ 0} neighborhood of the target ℂP^{3}, and is given by the formula:

[z_{0} : 1 : z_{2}] ↦ [z_{0} : 1 : z_{2} : \frac{t · Q(z_{0}, 1, z_{2})}{P(z_{0}, 1, z_{2})}] ,
(6.5)  (z_{0}, z_{2}) ↦ (z_{0}, z_{2}, \frac{t · (2z_{0}^{2} + 2z_{0}z̄_{2} − z_{0}z_{2} + 2z_{2}z̄_{0})}{6z_{0}z̄_{0} + 1 + 6z_{2}z̄_{2}}) .

This is another graph of a rational function G_{t}(z_{0}, z̄_{0}, z_{2}, z̄_{2}) = t · Q/P with a nonvanishing denominator, which is real analytic in z, z̄ and t. Since we have already found all the CR singularities of the form ι_{t}([1 : z_{1} : z_{2}]), we can simplify the computational problem of finding new CR singularities in the graph of G_{t} by adding the equation z_{0} = 0 to get this system:

(6.6)  \frac{d}{dz̄_{0}} G_{t} = 0, \frac{d}{dz̄_{2}} G_{t} = 0, z_{0} = 0.

The solution set does not depend on t (for t ≠ 0), and a calculation with [Maple] shows that (ζ_{0}, ζ_{2}) = (0, 0) is the only solution of (6.6). The graph of G_{t} is already in standard position; at t = 1, the quadratic part of the defining function is the numerator from (6.5):

G_{1}(z_{0}, z̄_{0}, z_{2}, z̄_{2}) = 2z_{0}^{2} + 2z_{0}z̄_{2} − z_{0}z_{2} + 2z_{2}z̄_{0} + O(3).

This is a CR singularity with index +1; the graph is in general position.

There is one last point to check, ι_{t}([0 : 0 : 1]) = [0 : 0 : 1 : 0]. The restriction of ι_{t} to the {z_{2} = 1} neighborhood is given by the formula:

[z_{0} : z_{1} : 1] ↦ [z_{0} : z_{1} : 1 : \frac{t · Q(z_{0}, z_{1}, 1)}{P(z_{0}, z_{1}, 1)}] ,
(z_{0}, z_{1}) ↦ (z_{0}, z_{1}, \frac{t · (2z_{0}^{2}z̄_{1} + 2z_{0}z_{1} − z_{0}z̄_{1} + 2z_{1}z̄_{0})}{6z_{0}z̄_{0} + z_{1}z̄_{1} + 6}) .

Since this image is also a real analytic graph, ι_{t} is a (global) real analytic embedding depending real analytically on t. The origin in this neighborhood is another CR singularity of the image; the graph is already in standard position and, for t ≠ 0, in general position, with a CR singular point of index +1.

The conclusion is that for t ≠ 0, ι_{t} : ℂP^{2} → ℂP^{3} has exactly seven CR singular points, at the image of

\{[1 : 0 : 2], [1 : ±\sqrt{3} : 1], [1 : ±3i : −1], [0 : 1 : 0], [0 : 0 : 1]\}.

Every image point is in N_{2}^{+} (positively oriented tangent space), with index +1, consistent with the characteristic class calculations of Example 3.7.

7. Summary

Theorem 7.1. Given a real analytic 4-dimensional submanifold M in ℂ^{3}, for any CR singular point there is a local holomorphic coordinate neighborhood so that the CR singular point is at the origin, the tangent space is the (z_{1}, z_{2})- hyperplane, and the local defining equation for M is given by

z_{3} = h(z, z̄)
(7.1) = (z̄_{1}, z̄_{2})N ( \begin{matrix}z_{1} \\ z_{2} \end{matrix} ) + Re ((z_{1}, z_{2})P ( \begin{matrix}z_{1} \\ z_{2} \end{matrix} )) + e(z_{1}, z̄_{1}, z_{2}, z̄_{2}),

where e(z, z̄) = O(3) is real analytic. The coefficient matrices N, P fall into one of the cases from Table 1, and exactly one (modulo equivalences as indicated).

In Table 1, the third column is the number of real moduli for each case. The last column indicates the sign of det(Γ) that can occur for various values of the entries of N and P: positive, negative, or zero (+, −, 0).

Theorem 7.2. Given a real analytic 4-dimensional submanifold M in ℂ^{3} with local defining equation (7.1) in one of the normal forms from Theorem 7.1, if the quadratic part of h(z, z̄) in (7.1) is real valued, then the coefficient matrices N, P fall into exactly one of the cases from the following Examples 7.3 – 7.7.

The whole numbers in the middle columns are invariants of the defining function h(z, z̄) under local biholomorphic coordinate changes that preserve the property of being in a quadratically flat normal form.

As previously mentioned, related lists of normal forms for pairs have appeared in [I], [E], and [CS].

Example 7.3. For N = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ), ρ(N) = 2, σ(N) = 2, see Table 2. The

diagonal elements of P in Table 2 are the previously mentioned “generalized Bishop invariants.” This is the only Example where cases with σ(Γ) = 4 occur, that is, where Γ is definite, or, equivalently, where the real Hessian is definite, as discussed in Subsection 4.1. These are the points called “flat elliptic” points by [DTZ_{1}], [DTZ_{2}], and [Dolbeault], and they appeared in the flat embedding of the ellipsoid in Example 6.1. The definiteness of Γ characterizes this elliptic property among all the equivalence classes from Theorem 7.2 and Examples 7.3 – 7.7. The first line in the above chart, where P = 0, represents the case considered by [HY].

We also see a difference from the real surface M ⊆ ℂ^{2} case, where a complex point (in N_{1}^{+}) has the elliptic property if and only if ind = +1. That characterization does not generalize to dimensions m = 4, n = 3; even in Table 2, flat elliptic complex points have ind = +1, but so do some quadratically flat, non-elliptic points.

Table 1. Normal forms for Theorem 7.1

\begin{matrix}N & P \\ \begin{matrix}( & \begin{matrix}1 & 0 \\ 0 & e^{iθ} \end{matrix} & ) \\ 0 < θ < π \end{matrix} & \begin{matrix}( \begin{matrix}a & b \\ b & d \end{matrix} ) & 5 & a > 0, d > 0, b ∼ −b ∈ ℂ & + − 0 \\ ( \begin{matrix}0 & b \\ b & d \end{matrix} ) & 3 & b ≥ 0, d ≥ 0 & + − 0 \\ ( \begin{matrix}a & b \\ b & 0 \end{matrix} ) & 3 & a > 0, b ≥ 0 & + − 0 \end{matrix} \\ ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) & ( \begin{matrix}a & 0 \\ 0 & d \end{matrix} ) 2 0 ≤ a ≤ d + − 0 \\ ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) & \begin{matrix}( \begin{matrix}a & 0 \\ 0 & d \end{matrix} ) & 2 & 0 ≤ a ≤ d & + − 0 \\ ( \begin{matrix}0 & b \\ b & 0 \end{matrix} ) & 1 & b > 0 & + \\ ( \begin{matrix}1 & 1 \\ 1 & 1 \end{matrix} ) & 0 & & + \end{matrix} \\ ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) & \begin{matrix}( \begin{matrix}0 & b \\ b & 1 \end{matrix} ) & 1 & b > 0 & + 0 \\ ( \begin{matrix}1 & 0 \\ 0 & d \end{matrix} ) & 2 & Im(d) > 0 & + \end{matrix} \\ \begin{matrix}( & \begin{matrix}0 & 1 \\ τ & 0 \end{matrix} & ) \\ 0 < τ < 1 \end{matrix} & \begin{matrix}( \begin{matrix}a & b \\ b & d \end{matrix} ) & 5 & b > 0, |a| = 1, (a, d) ∼ (−a, −d) & + − 0 \\ ( \begin{matrix}0 & b \\ b & d \end{matrix} ) & 3 & b > 0, |d| = 1, d ∼ −d & + − 0 \\ ( \begin{matrix}0 & b \\ b & 0 \end{matrix} ) & 2 & b > 0 & + − 0 \\ ( \begin{matrix}1 & 0 \\ 0 & d \end{matrix} ) & 3 & d ∈ ℂ & + 0 \\ ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) & 1 & & + \\ ( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} ) & 1 & & + \end{matrix} \\ ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) & \begin{matrix}( \begin{matrix}a & b \\ b & 1 \end{matrix} ) & 3 & b > 0, a ∈ ℂ & + − 0 \\ ( \begin{matrix}1 & b \\ b & 0 \end{matrix} ) & 1 & b > 0 & + − 0 \\ ( \begin{matrix}0 & b \\ b & 0 \end{matrix} ) & 1 & b > 0 & + − 0 \\ ( \begin{matrix}a & 0 \\ 0 & 1 \end{matrix} ) & 1 & a ≥ 0 & + 0 \\ ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) & 0 & & 0 \\ ( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} ) & 0 & & 0 \end{matrix} \end{matrix}

Table 1, continued

\begin{matrix}N & P \\ ( \begin{matrix}0 & 1 \\ 1 & i \end{matrix} ) & \begin{matrix}( \begin{matrix}a & 0 \\ 0 & d \end{matrix} ) & 3 & a > 0, d ∈ ℂ & + − 0 \\ ( \begin{matrix}0 & b \\ b & 0 \end{matrix} ) & 1 & b > 0 & + 0 \\ ( \begin{matrix}0 & 0 \\ 0 & d \end{matrix} ) & 1 & d ≥ 0 & + \end{matrix} \\ ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) & \begin{matrix}( \begin{matrix}a & 0 \\ 0 & 1 \end{matrix} ) & 1 & a ≥ 0 & + − 0 \\ ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) & 0 & & + \\ ( \begin{matrix}a & 0 \\ 0 & 0 \end{matrix} ) & 1 & a ≥ 0 & 0 \end{matrix} \\ ( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} ) & \begin{matrix}( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) & 0 & + \\ ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) & 0 & 0 \\ ( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} ) & 0 & 0 \end{matrix} \end{matrix} .

The normal forms in Table 2 with full rank ρ(Γ) = 4 and indefinite signature σ(Γ) = 2 or 0 represent the equivalence classes of points called “hyperbolic” by [Dolbeault].

Example 7.4. For N = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ), ρ(N) = 2, σ(N) = 0, see Table 3. In

Table 3, we see that the discrete invariants in the last 5 columns are repeated in a few cases, so they are not enough to distinguish inequivalent matrix normal forms of different shapes (such as diagonalizable or not).

Example 7.5. For N = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ), ρ(N) = 2, σ(N) = 0, see Table 4. Some

of the rows in Table 4 have the same ρ and σ data as rows from the previous Example.

Example 7.6. For N = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ), ρ(N) = 1, σ(N) = 1, see Table 5.

Example 7.7. For N = ( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} ), ρ(N) = 0, σ(N) = 0, see Table 6.

Table 2. Normal forms for Theorem 7.2, Example 7.3

P  ρ(P) ρ(N|P) ρ(Γ) σ(Γ) sign(det(Γ)) ( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} )  0  2  4  4  + ( \begin{matrix}0 & 0 \\ 0 & d \end{matrix} ) 0 < d < 1  1  2  4  4  + ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} )  1  2  3  3  0 ( \begin{matrix}0 & 0 \\ 0 & d \end{matrix} )  1 < d  1  2  4  2  − ( \begin{matrix}a & 0 \\ 0 & d \end{matrix} ) 0 < a ≤ d < 1 2  2  4  4  + ( \begin{matrix}a & 0 \\ 0 & 1 \end{matrix} ) 0 < a < 1  2  2  3  3  0 ( \begin{matrix}a & 0 \\ 0 & d \end{matrix} ) 0 < a < 1 < d 2  2  4  2  − ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} )  2  2  2  2  0 ( \begin{matrix}1 & 0 \\ 0 & d \end{matrix} )  1 < d  2  2  3  1  0 ( \begin{matrix}a & 0 \\ 0 & d \end{matrix} ) 1 < a ≤ d  2  2  4  0  + .

Acknowledgments. The author was motivated to start writing this paper after conversations with F. Forstnerič and D. Zaitsev, during an overseas trip funded in part by the Indiana University Office of International Programs. In particular, Subsection 2.4.3 addresses a question posed by Zaitsev.

References

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Table 3. Normal forms for Theorem 7.2, Example 7.4

P  ρ(P) ρ(N|P) ρ(Γ) σ(Γ) sign(det(Γ)) ( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} )  0  2  4  0  + ( \begin{matrix}0 & 0 \\ 0 & d \end{matrix} ) 0 < d < 1  1  2  4  0  + ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} )  1  2  3  1  0 ( \begin{matrix}0 & 0 \\ 0 & d \end{matrix} )  1 < d  1  2  4  2  − ( \begin{matrix}a & 0 \\ 0 & d \end{matrix} ) 0 < a ≤ d < 1 2  2  4  0  + ( \begin{matrix}a & 0 \\ 0 & 1 \end{matrix} ) 0 < a < 1  2  2  3  1  0 ( \begin{matrix}a & 0 \\ 0 & d \end{matrix} ) 0 < a < 1 < d 2  2  4  2  − ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} )  2  2  2  0  0 ( \begin{matrix}1 & 0 \\ 0 & d \end{matrix} )  1 < d  2  2  3  1  0 ( \begin{matrix}a & 0 \\ 0 & d \end{matrix} ) 1 < a ≤ d  2  2  4  0  + ( \begin{matrix}1 & 1 \\ 1 & 1 \end{matrix} )  1  2  4  0  + ( \begin{matrix}0 & b \\ b & 0 \end{matrix} )  0 < b  2  2  4  0  + .

Table 4. Normal forms for Theorem 7.2, Example 7.5

P  ρ(P) ρ(N|P) ρ(Γ) σ(Γ) sign(det(Γ)) ( \begin{matrix}0 & b \\ b & 1 \end{matrix} ) 0 < b ≠ 1 2  2  4  0  + ( \begin{matrix}0 & 1 \\ 1 & 1 \end{matrix} )  2  2  3  1  0 ( \begin{matrix}1 & 0 \\ 0 & d \end{matrix} ) 0 < Im(d) 2  2  4  0  + .

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Table 5. Normal forms for Theorem 7.2, Example 7.6

P  ρ(P) ρ(N|P) ρ(Γ) σ(Γ) sign(det(Γ)) ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} )  1  2  4  2  − ( \begin{matrix}a & 0 \\ 0 & 1 \end{matrix} ) 0 < a < 1 2  2  4  2  − ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} )  2  2  3  1  0 ( \begin{matrix}a & 0 \\ 0 & 1 \end{matrix} ) 1 < a  2  2  4  0  + ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )  2  2  4  0  + ( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} )  0  1  2  2  0 ( \begin{matrix}a & 0 \\ 0 & 0 \end{matrix} ) 0 < a < 1 1  1  2  2  0 ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} )  1  1  1  1  0 ( \begin{matrix}a & 0 \\ 0 & 0 \end{matrix} ) 1 < a  1  1  2  0  0 .

Table 6. Normal forms for Theorem 7.2, Example 7.7

P  ρ(P) ρ(N|P) ρ(Γ) σ(Γ) sign(det(Γ)) ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} )  2  2  4  0  + ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} )  1  1  2  0  0 ( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} )  0  0  0  0  0 .

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

E-mail address: CoffmanA@ipfw.edu