Special Normal Form of a hyperbolic CR manifold in ℂ^{4} ^{∗}^{†}

Vladimir V. Ežov ^{‡} University of Adelaide Department of Pure Mathematics Adelaide, SA 5005, South Australia vezhov@maths.adelaide.edu.au

Gerd Schmalz ^{‡‡} Mathematisches Institut der Universität Bonn Beringstraße 1 D-53115 Bonn schmalz@uni-bonn.de

June 18, 1998

Abstract In this paper we give a special normal form for a non-semiquadratic

hyperbolic CR-manifold M of codimension 2 in ℂ^{4}, i.e., a construction of coordinates where the equation of M satisfies certain conditions. The coordinates are determined up to a linear coordinate change.

∗Mathematics Subject Classification: primary 32C16; secondary 32C05, 32H02 †Keywords and Phrases: CR-manifolds, invariants, classification ‡Research of the first author was supported by ARC and SFB 256. ‡‡Research of the second author was supported by SFB 256 of Deutsche Forschungsge-

meinschaft.

1 Introduction

In 1907 Poincaré [Poi07] posed the problem of local biholomorphic equivalence of two real-analytic hypersurfaces in ℂ^{2}. It is interesting to consider this problem for real hypersurfaces in ℂ^{N} or, more generally for real-analytic CR-submanifolds in ℂ^{N}. One approach to this problem, that is due to Chern and Moser [CM74], is the construction of normal forms of the hypersurfaces, i.e., special coordinate systems with the property that certain monomials in the equation of the hypersurface, considered as a power series, vanish. This construction is universal for all hypersurfaces with a given signature of the Levi-form. The normal form condition determines the coordinate system up to a rather complicated action of the finite-dimensional isotropy group of the quadric of maximal osculation. This group is covered by SU(n − p, p + 1), where n = N − 1 and p is the signature of the Levi-form. If both hypersurfaces are given in normal form the equivalence problem reduces to the (still complicated) problem to find a transformation out of a finite-dimensional variety that maps one hypersurface to the other.

Webster introduced in 1978 [Web78] a special normal form that takes into account higher order terms of a given hypersurface. By imposing additional requirements he reduces the freedom in the choice of normal coordinates to a subgroup of GL(n, ℂ) that acts by linear transformations.

In this paper we construct a special normal form for hyperbolic CRmanifolds of codimension 2 in ℂ^{4}. By hyperbolic CR-manifold we mean a CR-submanifold M ⊂ ℂ^{4} that can be written in the form

v_{1} = |z_{1}|^{2} + N_{1}(z, z̄, u)  (1)
v_{2} = |z_{2}|^{2} + N_{2}(z, z̄, u),

where z_{1}, z_{2}, w_{1}, w_{2} are coordinates in ℂ^{4} and w_{j} = u_{j} + iv_{j} and N_{j} does not contain any terms of order less than 3. Hyperbolic CR-manifolds in C^{4} can also be characterised as the strictly pseudo-convex CR-manifolds of codimension 2 in ℂ^{4}. For such manifolds Loboda constructed a normal form that is analogous to Chern-Moser’s normal form for hypersurfaces. Loboda’s normal form conditions can be described as follows:

Set

N_{j}(z, z̄, u) = N_{j}^{1}(z_{j}, z̄_{j}, u_{j}) + N_{j}^{2}(z, z̄, u),

where N_{j}^{1}(z_{j}, z̄_{j}, u_{j}) consists of all monomials that depend only on z_{j}, z̄_{j}, u_{j}. Then

N_{j}^{1} = 2 Re(h_{42}^{j}(u)z_{j}^{4}z̄_{j}^{2}) + ∑_{\begin{matrix}k, l ≥2 \\ k+l≥7 \end{matrix}} h_{kl}^{j}(u)z_{j}^{k}z̄_{j}^{l},

where h_{kl}^{j}(u) are real-analytic functions of u. Notice that these are exactly Chern-Moser’s conditions for the normal form of a hypersurface in ℂ^{2}.

For the terms N_{2}^{j} the following holds

(N_{2}^{j})_{k0} ≡ 0, for k ≥ 1  (N_{2}^{j})_{11} ≡ 0
\frac{∂}{∂z̄_{j}} (N_{2}^{j})_{k1} ≡ 0, for k ≥ 2  \frac{∂^{2}}{∂z_{1}∂z_{2}} (N_{2}^{j})_{21} ≡ 0
\frac{∂^{4}}{∂z_{1}∂z_{2}∂z̄_{1}∂z̄_{2}} (N_{2}^{j})_{22} ≡ 0

By N_{0}(M) we denote the family of biholomorphic, origin preserving transformations that map M (in normal form) into M′ that has also normal form. If Φ ∈ N_{0}(M) : z′ = f(z, w), w′ = g(z, w) then

L_{C} : z′ = Cz,  w′ = ρw  (2)

with

C = (\frac{∂f}{∂z} )|_{0} , ρ = (\frac{∂g}{∂w})|_{0}

also belongs to N_{0}(M). Moreover, C and ρ must then have the form

C = (\begin{matrix}λ_{1}e^{iψ_{1}} & 0 \\ 0 & λ_{2}e^{iψ_{2}} \end{matrix}) , ρ = (\begin{matrix}λ_{1}^{2} & 0 \\ 0 & λ_{2}^{2} \end{matrix}) ,

or,

C = (\begin{matrix}0 & λ_{1}e^{iψ_{1}} \\ λ_{2}e^{iψ_{2}} & 0 \end{matrix}) , ρ = (\begin{matrix}0 & λ_{1}^{2} \\ λ_{2}^{2} & 0 \end{matrix}) ,

where λ_{j} > 0, ψ_{j} ∈ ℝ. The group of all (C, ρ)- transformations will be denoted by GL(Q).

Thus, any Ψ ∈ N_{0}(M) can be decomposed into Ψ = L_{C} ∘ Ψ_{id} with Ψ_{id} ∈ N_{0,id}(M), where

N_{0,id}(M) = \{Φ ∈ N_{0}(M) : (\frac{∂f}{∂z} )|_{0} = id\} .

It follows from Loboda’s theorem that any Φ ∈ N_{0,id}(M) is uniquely determined by the data

a_{j} := \frac{∂f_{j}}{∂w_{j}} |_{0} , r_{j} := \frac{1}{2} \frac{∂^{2}g_{j}}{(∂w_{j})^{2}} |_{0} .

If M is the hyperbolic quadric

M : v_{1} = |z_{1}|^{2}, v_{2} = |z_{2}|^{2}

then for any possible initial data there exists an automorphism, hence any normalisation is automatically an automorphism.

For arbitrary M the right hand side of equation (1) can be expanded into a series of homogeneous polynomials where the weighted degree of each monomial is the sum of the degree with respect to z and z̄ and twice the degree with respect to u. If M is non-quadric and given in normal form then there is a smallest weight γ such that the homogeneous component N_{γ} with weight γ of N = (N_{1}, N_{2}) is different from 0. We will refer to this N_{γ} as to the first non-quadratic term. It can be easily verified that any normalisation Φ ∈ N_{0}(M) is equal to the quadric automorphism with the same initial data up to terms of weighted degree ≥ γ (For more elaborate explication see [Sch95]).

Among the non-quadratic hyperbolic manifolds there is a special type of manifolds that are called semi-quadratic manifolds and should be separately dealt with. A semi-quadratic manifolds is given by the equations

v_{1} = |z_{1}|^{2}
v_{2} = |z_{2}|^{2} + N^{2}(z_{2}, z̄_{2}, u_{2}),

where z_{1}, z_{2}, w_{1}, w_{2} are normal coordinates, i.e., they are a Cartesian product of a sphere in ℂ^{2} and a non-spherical hypersurface in ℂ^{2} that is given in Chern-Moser’s normal form.

The construction of a special normal form reduces then to Chern-Moser-Webster’s special normal form for hypersurfaces. A coordinate change from one special normal form to another is the a fractional linear sphere automorphism in the variables z_{1}, w_{1} and a linear transformation in the variables z_{2}, w_{2}.

Now, we may restrict ourselves to the case of non-semiquadratic manifolds. The construction and the condition of the special normal form for a given non-semiquadratic manifold M depends on the concrete form of the first non-quadratic, resp. non-semiquadratic, term. We will consider 4 different cases of non-semiquadratic manifolds that appeared in [Sch95] where it was proved that for such manifolds there exist normal coordinates where all automorphisms are linear. This is, of course, necessary for a construction of special normal forms with the property that passing to another special normal coordinates is provided by linear transformations because any automorphism is particularly a transformation to other special normal coordinates.

In Section 2 we recall the rough scheme of the proof of the following theorem from [Sch95]:

Theorem 1 Let M be a non-semiquadratic, real-analytic CR-manifold in ℂ^{4}. Then any isotropic automorphism Φ = (f, g), considered as a normalisation, is uniquely determined by the parameter

C = \frac{∂f}{∂z} |_{0}

Section 3 is devoted to the description of the special normal form and to the proof of the existence and uniqueness (up to linear coordinate changes) special coordinates.

We remark that a special normal form for non-umbilic hyperbolic CR manifolds of codimension 2 in ℂ^{4} was constructed by Loboda in [Lob90].

2 Sketch of the proof of Theorem 1

The proof of Theorem 1 is very long and very technical. Here we will only sketch the main ideas.

Step 1. One has to show that any automorphism with C = id is trivial. Any such automorphism can be embedded in some one-parametric subgroup Φ_{t}. From

Im g_{t}(z, w) − ⟨f_{t}(z, w), f_{t}(z, w)⟩ − N(f_{t}, f̄_{t}, Re g_{t})|_{Im w=⟨z,z⟩+N} ≡ 0 (3)

one obtains that the derivative with respect to t for t = 0 must vanish. Denote this function by P (z, z̄, u).

Step 2. The function P (z, z̄, u) can be expressed by

P (z, z̄, u) = 2 Re χ(v − ⟨z, z⟩ − N(z, z̄, u))|_{Im w=⟨z,z⟩+N} ,

where χ is a vector field that is decomposed into a series of homogeneous components with respect to the weight

χ = χ_{1} + χ_{2} + χ_{γ−1} + ⋯

Here, one has to take into account that the basis vectors \frac{∂}{∂z_{j}}, resp., \frac{∂}{∂w_{j}} have the weight −1, resp., −2. Then the index indicates the weight of the component. The vector fields χ_{1} and χ_{2} do not depend on M and have the explicit form

χ_{1} = ∑_{j=1,2} (a_{j}w_{j} + 2iā_{j}(z_{j})^{2}) \frac{∂}{∂z_{j}} + 2iā_{j}z_{j}w_{j} \frac{∂}{∂w_{j}}

χ_{2} = ∑_{j=1,2} r_{j}z_{j}w_{j} \frac{∂}{∂z_{j}} + r_{j}(w_{j})^{2} \frac{∂}{∂w_{j}} .

Step 3. Besides of the weighted degree, there is another characteristic of a monomial that plays an important rôle in the proof. This characteristic is called j-defect (for j = 1, 2) and is defined as the difference of the degree in z_{j} and in z̄_{j}.

Now, suppose that a_{j} ≠ 0. Without loss of generality one can then assume that a_{1} = i. One has to show that then equation (3) cannot be satisfied. This implies that there are no isotropic automorphisms Φ of M with Φ ∈ Aut_{0,id}. We distinguish four different cases:

1. M has the form

v_{1} = |z_{1}|^{2} + N^{1}(z, z̄, u)
v_{2} = |z_{2}|^{2} + N^{2}(z, z̄, u),  (4)

where N_{γ_{1},δ_{1}}^{1} ≢ 0 is the term of maximal 1-defect δ_{1} + 1 of the first nonvanishing term of N^{1}. Then the term in P^{1} of weight γ_{1} and 1-defect δ_{1} takes the form

P_{γ_{1}+1,δ_{1}+1}^{1} = χ_{(1)}|_{v_{1}=|z_{1}|^{2}} N_{γ_{1},δ_{1}}^{1} −2z_{1}N_{γ_{1},δ_{1}}^{1} + χ_{γ_{1}−1,δ_{1}+1}|_{v_{1}=|z_{1}|^{2}} (|z_{1}|^{2}−v_{1}),

where

χ_{(j)} = iā_{j} (2(z_{j})^{2} \frac{∂}{∂z_{j}} + 2(u_{j} + iv_{j})z_{j} \frac{∂}{∂w_{j}} − i(u_{j} − iv_{j})\frac{∂}{∂z̄_{j}} )

is the part of the vector field χ_{1} that increases the j-defect by 1.

Due to (3), this expression must vanish. For the first case we suppose that the sum the terms including N^{1} is different from 0. Then they must be compensated by the last term that includes χ_{γ_{1}−1,δ_{1}+1}. This turns out to be impossible because then the last term has non-normal components that cannot occur in the first two terms. Hence, a_{1} ≠ 0 implies that the polynomials P_{γ_{1}+1,δ_{1}+1}^{1} cannot vanish, i.e., for a_{1} running over ℂ they form a one-dimensional subspace in the space of polynomials with weight γ_{1} + 1 and j-defect δ_{1} + 1.

2. For the second case we assume that M has the form

v_{1} = |z_{1}|^{2}
v_{2} = |z_{2}|^{2} + N^{2}(z, z̄, u),  (5)

where N_{γ_{2},δ_{2}}^{2} ≢ 0 is the first non-vanishing term with maximal 2-defect, N_{γ_{1},δ_{1}}^{2} is the first term that depends on one of the variables z_{1}, z̄_{1}, u_{1} with weight γ_{1} and maximal 1-defect δ_{1}. Then a_{1} ≠ 0 implies P_{γ_{1}+1,δ_{1}+1}^{2} ≠ 0 (even without the assumption N^{1} ≡ 0). Furthermore, the second case shall be characterised by a_{2} ≠ 0 implies P_{γ_{2}+1,δ_{2}+1}^{2} ≠ 0, as above. In this case the polynomials P_{γ_{j}+1,δ_{j}+1}^{j} form one-dimensional subspaces in the spaces of polynomials with weight γ_{j} + 1 and j-defect δ_{j} + 1 for a_{j} running over ℂ.

3. For the third case M has the form (4) but a_{1} ≠ 0 does not imply that P_{γ_{1}+1}^{1} ≠ 0. Then one obtains that N_{γ_{1}}^{1} must have the special form

v_{1} = |z_{1}|^{2} + (u_{1} + i|z_{1}|^{2})(u_{1} − i|z_{1}|^{2})Π_{2}(z_{2}, z̄_{2}, u_{2}) + ⋯,

where Π_{2} is not constant and depends only on the indicated variables. Then it follows from a_{2} ≠ 0 that P_{γ_{1}+1}^{1} ≠ 0. There are two possibilities for N^{2}: either there is a first component N_{γ_{2}}^{2} that depends on one of the variables z_{1}, z̄_{1}, u_{1} and a_{1} ≠ 0 implies P_{γ_{2}+1}^{2} ≠ 0, or N^{2} depends only on z_{2}, z̄_{2}, u_{2}. The first alternative characterises the third case. Then one finds out that a_{j} running over ℂ produces one-dimensional subspaces of P_{γ_{j}+1,δ_{j}+1}^{l} (j ≠ l) in the space of polynomials with weight γ_{j} + 1 and j-defect δ_{j} + 1.

4. The second alternative from above constitutes the fourth case. Then there exists a number k with 2 ≤ k ≤ γ such that P_{γ_{1}+k}^{1} ≠ 0. We will call M exceptional of order k.

Step 4. If r_{j} ≠ 0 then P_{γ+2} ≠ 0.

3 The special normal form

Consider the vector fields

χ_{(j)} = iā_{j} (2(z_{j})^{2} \frac{∂}{∂z_{j}} + 2(u_{j} + iv_{j})z_{j} \frac{∂}{∂w_{j}} − i(u_{j} − iv_{j})\frac{∂}{∂z̄_{j}} )  (6)
χ_{2}^{j} = r_{j}w_{j}z_{j} \frac{∂}{∂z_{j}} + r_{j}(w_{j})^{2} \frac{∂}{∂w_{j}}  (7)
χ_{γ} = f_{γ+1}(z, w)\frac{∂}{∂z} + g_{γ+2}(z, w)\frac{∂}{∂w},  γ ≥ 3,  (8)

with a_{j} ∈ ℂ, r_{j} ∈ ℝ. Notice that χ_{(j)} increases the j-defect of a monomial by 1 and that

2 Re χ_{1} = χ_{(1)} + χ_{(2)} + \overline{χ_{(1)}} + \overline{χ_{(2)}}
χ_{2} = χ_{2}^{1} + χ_{2}^{2}.

By χ_{γ,δ_{j}} resp. χ_{γ,d} we will denote the component of j-defect δ_{j} resp. of total defect d = δ_{1} + δ_{2} in χ_{γ}.

We distinguish now four types of manifolds in dependence on the first nonquadratic terms in the expansion of the equation in normal form. Since the first order term of any renormalisation is is a linear mapping L_{C} as described above, the dependence of the first non-quadratic terms in the equation of M depends in a simple way from the chosen normal coordinates: only the coefficients at the monomials change. Therefore, the definitions below are correct.

These four types of manifolds are according to the four cases in Step 3 of the proof above.

1. Both N^{j} ≢ 0 and have not the special form indicated in case 3. Let N_{γ_{j}}^{j} be the first non-quadratic terms. Collect in N_{γ_{j}}^{j} the terms of maximal total defect d_{j}, then within this polynomial the terms of maximal jdefect δ_{j} and then the terms of maximal weight ω_{j} with respect to (z_{j}, z̄_{j}, u_{j}). Finally, choose in the resulting expression a polynomial of the form

N̂_{γ_{j},d_{j}}^{j} = Ñ_{ω_{j},δ_{j}}^{j}(z_{j}, z̄_{j}, u_{j})Π^{j},

where Π^{j} is a monomial in z_{l}, z̄_{l}, u_{l} (l ≠ j) of minimal degree with respect to z_{l}, z̄_{l}. Consider space

P^{j} = {P_{γ_{j}+1,δ_{j}+1}^{j} = χ_{(j)}N̂_{γ_{j},δ_{j}}^{j} − 2z_{j}N̂_{γ_{j},δ_{j}}^{j} + χ_{γ_{j}−1,δ_{j}+1}(|z_{j}|^{2} − v_{j})|_{v_{j}=|z_{j}|^{2}}} , (9)

where χ_{γ_{j}−1,δ_{j}+1} is uniquely determined for given a_{j} by the condition that P^{j} consists of normal terms. The arguments from [Sch95] (cf. Section 2) applied to normalisations (instead of automorphisms) show that these spaces are 1-dimensional complex spaces for a_{j} running over ℂ. We can make use of these arguments because the action of automorphisms and normalisations on the first non-quadratic terms are the same.

Analogously, we define the 1-dimensional real spaces

Q^{j} = {Q_{γ_{j}+2,δ_{j}}^{j} = χ_{2}^{j}N̂_{γ_{j},d_{j}}^{j} |_{v_{j}=|z_{j}|^{2}}} .  (10)

2. One of the N^{j} ≡ 0, say N^{1} ≡ 0. Since M is not semi-quadratic, N^{2} must then depend on one of the z_{1}, z̄_{1}, u_{1}. We define N̂^{2} in the same way as above. N̂^{1} will be defined analogously, but starting from N_{γ_{1},δ_{1}}^{2} the term in N^{2} of minimal weight γ_{1} and maximal 1-defect δ_{1} that depends on one of the variables z_{1}, z̄_{1}, u_{1}. Now, we define by (9), resp. (10), the 1-dimensional complex spaces P^{j}, resp. the 1-dimensional real spaces Q^{j}.

3. One of the N^{j}, has the special form from case 3. and, therefore, must depend on one of the variables z_{l}, z̄_{l}, u_{l} (l ≠ j), and N^{l} depends on z_{j}, z̄_{j}, u_{j}. N̂^{j} is then derived from N_{γ_{j},δ_{j}}^{l}, (l ≠ j), the component of minimal weight and maximal j-defect that contains one of the variables z_{j}, z̄_{j}, u_{j}.

The spaces P^{j} and Q^{j} will also be defined by (9) and (10). According to [Sch95], P^{j} are complex 1-dimensional for a_{j} running over ℂ and the Q^{j} are real 1-dimensional for r_{j} running over ℝ.

4. One of the N^{j}, say N^{1} has the special form from case 3., and N^{2} does not depend on z_{1}, z̄_{1}, u_{1}. Then there exist an integer k such that the spaces generated by P_{γ_{1}+k,1}^{1}, resp., P^{1}γ_{1} + 1, δ_{2} + 1 for a_{1}, resp., a_{2} in ℂ are one-dimensional. In this case N̂^{2}, P^{2} and Q^{j} are defined as for manifolds of type 3. N̂^{1} will be derived from N_{γ_{1},δ_{1}}^{1}, (γ_{1} = γ_{2} + k − 1), the term of minimal weight and maximal 1-defect that produces a a one-dimensional space of polynomials P_{γ_{2}+k,1}.

Set

P^{1} = {P_{γ_{1}+k,δ_{1}+1}^{1} = χ_{(j)}N̂_{γ_{j},δ_{j}}^{1} − 2z_{1}N̂_{γ_{1}+k−1,δ_{1}}^{1}

+ χ_{γ_{1}+k−2,δ_{1}+1}(|z_{1}|^{2} − v_{1})|_{v_{1}=|z_{1}|^{2}} }.

In the spaces of polynomials with given weight we introduce an inner product such that the monomials form an orthonormal basis. We will use this inner product in order to choose complementary subspaces.

Now we are ready to give the definition of the s.n.f.

Definition 1 A non-semiquadratic CR-manifold M has s.n.f. if for some choice of N̂^{j} the following conditions are satisfied:

If M is of type 1, then

N_{γ_{j}+1,δ_{j}+1}^{j} ∈ P^{j}^{⊥}, and N_{γ_{j}+2,δ_{j}}^{j} ∈ Q^{j}^{⊥}.

If M is of type 2, then

N_{γ_{j}+1,δ_{j}+1}^{2} ∈ P^{j}^{⊥}, and N_{γ_{j}+2,δ_{j}}^{2} ∈ Q^{j}^{⊥}.

For M of type 3

N_{γ_{j}+1,δ_{j}+1}^{l} ∈ P^{j}^{⊥}, and N_{γ_{j}+2,δ_{j}}^{l} ∈ Q^{j}^{⊥}, (l ≠ j),

and for M of exceptional type

N_{γ_{1}+k,δ_{1}+1}^{1} ∈ P^{1}^{⊥}, N_{γ_{1}+1,δ_{2}+1}^{1} ∈ P^{2}^{⊥}, and N_{γ_{j}+2,δ_{j}}^{j} ∈ Q^{j}^{⊥}.

By definition M has also s.n.f. if its equation can be obtained from one of the equations above by interchanging z_{1} with z_{2} and w_{1} with w_{2}.

Theorem 2 If M is a hyperbolic non-semiquadratic CR-manifold in ℂ^{4} of codimension 2 then there exist coordinates such that the equation of M has special normal form (s.n.f.). The coordinates where M has s.n.f. are determined up to a linear transformation of the form (2).

Proof. Let M be given in normal form in sense of Loboda and let Φ_{a} = (f_{a}, g_{a}) be a normalisation of M with parameters (C = id, a = (a_{1}, a_{2}), r = 0). The expansion of this normalisation into weighted homogeneous components differs from a quadric automorphism only starting from term of weight γ in f_{a} and of weight γ + 1 in g_{a}. The spaces P^{j} were chosen so that for suitable a_{1}, a_{2} the P^{j}-component of the corresponding term in the new equation can be made equal to zero.

After performing a normalisation with parameters (C = id, a = 0, r = (r_{1}, r_{2})) with suitable r_{j} the condition that involves the Q^{j} will be also satisfied.

Now we turn to the proof of the uniqueness statement. By inspection one easily verifies that any linear mapping of the form (2) preserves the s.n.f.

Now, let Φ be a local coordinate change such that the new equation has also s.n.f. Then Φ is a renormalisation and its linear part is equal to some L_{C}. The mapping L_{C}^{−1} ∘ Φ is also a coordinate change that takes M to s.n.f. and its linear part is the identity. Such renormalisations act on the first nonquadratic terms like quadric automorphisms and, therefore, would produce non-normal terms for one of the a_{j}, r_{j} being different from 0. This shows that a_{j} = r_{j} = 0. But, according to Loboda’s theorem, then L_{C}^{−1} ∘ Φ = id, i.e., Φ = L_{C}. □

References

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