Canonical form of the 4th order polynomial in the normal equation of a real hypersurface in

ℂ^{3}

Vladimir V. Ežov ^{∗} University of Adelaide

Department of Pure Mathematics Adelaide, SA 5005, South Australia

vezhov@maths.adelaide.edu.au

Aleksandr V. Loboda ^{†}

Voronezh State Academy of Architecture and Construction

Voronezh, 20-let Oktyabrya 84, Russia, 394006 lob@vgasa.voronezh.su

Gerd Schmalz^{‡}

Mathematisches Institut der Universität Bonn,

Beringstraße 1, D-53115 Bonn schmalz@uni-bonn.de

June 25, 1998

Abstract

We classify the first non-quadratic term in the Chern-Moser normal form of a real hypersurface in ℂ^{3}. Our classification is based on the study of two real forms of the 5-dimensional irreducible representation of the group SL(2, ℂ). This result can be applied to the problem of classification of homogeneous real hypersurfaces by means of the normal equation.

∗Research of the first author was supported by ARC and MPI Bonn.

†Research of the second author was supported by SFB 256 of DFG.

‡Research of the third author was supported by SFB 256 of DFG.

Let M ⊂ ℂ^{3} be a smooth real hypersurface with non-degenerate Levi form. According to the results by Chern and Moser [CM74], there exist coordinates z_{1}, z_{2}, w = u + iv such that the equation of M takes the form

v = ⟨z, z⟩ + F_{22} + O((|z| + |u|)^{5}),  (1)

where ⟨z, z⟩ is a hermitian form and F_{22} is a polynomial of degree 2 in z, and of degree 2 in z̄. Thus, up to order three, any hypersurface with non-degenerate Levi form is equivalent to the quadric

v = ⟨z, z⟩.

This means that F_{22} is the lowest order term that can be different in the normalised equations for two hypersurfaces with given Levi form. In dependence of the signature of the Levi form we distinguish two cases

⟨z, z⟩_{+} = |z_{1}|^{2} + |z_{2}|^{2}  pseudo-convex case
⟨z, z⟩_{−} = 2 Im z_{1}z̄_{2}  indefinite case.

The polynomial F_{22} satisfies the “harmonicity”condition ΔF_{22} = 0, where

Δ_{+} = \frac{∂^{2}}{∂z_{1}∂z̄_{1}} + \frac{∂^{2}}{∂z_{2}∂z̄_{2}}  in the pseudo-convex case
Δ_{−} = \frac{∂^{2}}{∂z_{1}∂z̄_{2}} − \frac{∂^{2}}{∂z_{2}∂z̄_{1}}  in the indefinite case.

Below we study the question to what form one can reduce the polynomial F_{22} by means of biholomorphic transformations that preserve the origin and respect the normal form of the equation of M.

We prove the following result.

Theorem 1 In the case of an indefinite Levi form the polynomial F_{22}(z, z̄) in the normalised equation (1) can be reduced to one of the following:

1. σ_{1} + aσ_{0} + σ_{−1} with a ≥ 0,

2. σ_{2} + aσ_{0} + σ_{−2} with a ≥ 2,

3. σ_{2} + σ_{0},

4. σ_{2},

5. σ_{1},

6. σ_{0} where

σ_{−2} = |z_{2}|^{4}, σ_{−1} = Re(z_{1}z̄_{2})|z_{2}|^{2}, σ_{0} = \frac{2}{3}|z_{1}|^{2}|z_{2}|^{2} + \frac{1}{3} Re z_{1}^{2}z̄_{2}^{2}

σ_{1} = Re(z_{1}z̄_{2})|z_{1}|^{2}, σ_{2} = |z_{1}|^{4}.

In the pseudo-convex case the polynomial F_{22}(z, z̄) in the normalised equation (1) can be reduced to one of the following:

1. ±(η_{−2} + iaη_{−1} + 2σ_{0} − iaη_{1} + η_{2}) with 0 ≤ a ≤ 1,

2. ±(η_{−2} − 2η_{−1} + 2σ_{0} − 2η_{1} + η_{2}),

3. ±i(η_{−2} + 2η_{−1} + 2η_{1} + η_{2}),

4. ±(η_{−2} + 4η_{−1} + 6σ_{0} + 4η_{1} + η_{2}),

5. ±(η_{−1} + aη_{0} + 2η_{1}) with a ≥ 0,

6. ±η_{0}, where

η_{−2} = η̄_{2} = z_{1}^{2}z̄_{2}^{2}, η_{−1} = η̄_{1} = (z_{1}z_{2}z̄_{2}^{2} − z_{1}^{2}z̄_{1}z̄_{2}), η_{0} = |z_{1}|^{4} − 4|z_{1}|^{2}|z_{2}|^{2} + |z_{2}|^{4}.

Proof. According to Chern Moser’s theorem, any biholomorphic mapping Φ that preserves the origin and respects the normal form conditions equals, up to 4-th order terms, to a fractional-linear automorphism of the quadric

z^{∗} = λ e^{iθ} J^{ε}U \frac{z + aw}{1 − 2i⟨z, a⟩ − (r + i⟨a, a⟩)w}
w^{∗} = (−1)^{ε}λ^{2} \frac{w}{1 − 2i⟨z, a⟩ − (r + i⟨a, a⟩)w},

where λ ∈ ℝ^{+}, θ ∈ [0, 2π), a ∈ ℂ^{2}, r ∈ ℝ. The matrix U is unitary, resp. pseudo-unitary, and unimodular,

J = (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) ,

and the parameter ε is always 0 in the pseudo-convex case and is 0 or 1 in the indefinite case.

Only the linear part of the mapping Φ effects the polynomial

F_{22}(z, z̄) = (−1)^{ε}λ^{2}F_{22}^{∗}(Uz, \overline{Uz}).

Thus, we have to study a linear action of the group of matrices C = λ e^{iθ} U at the space of harmonic polynomials of bidegree (2, 2). This decomposition is unique up to a multiplier −1 if one takes U unimodular. One sees at once that e^{iθ} does not effect F_{22} at all and that λ acts by simple multiplication with λ^{2}.

Consider the indefinite case. Then U is from SL(2, ℝ), i.e., we have to describe an SL(2, ℝ) action at the 5-dimensional space F_{22} of harmonic polynomials of bidegree (2, 2) spanned by σ_{j} j = −2, …, 2. In order to do this, we pass to the isomorphic 5-dimensional standard representation of SL(2, ℝ) at the space P_{5} = ℝ^{(4)}[X : Y] of real homogeneous polynomials of degree 4 in X, Y. The isomorphism is established by the assignment

σ_{j} ↦ σ̂_{j} = X^{2−j}Y^{2+j}.

Since SL(2, ℝ) is connected it suffices to check that the mapping is equivariant with respect to the corresponding 𝔰𝔩 (2, ℝ) actions τ : 𝔰𝔩 (2, ℝ) → 𝔤𝔩 (F_{22}), resp. τ̂ : 𝔰𝔩 (2, ℝ) → 𝔤𝔩 (P_{5}). We have then

τ(T_{−}) = 2 Re z_{2} \frac{∂}{∂z_{1}} ,  τ(T_{+}) = 2 Re z_{1} \frac{∂}{∂z_{2}} , τ(T_{0}) = 2 Re (z_{1} \frac{∂}{∂z_{1}} + z_{2} \frac{∂}{∂z_{2}} ) ,
τ̂(T_{−}) = X \frac{∂}{∂Y} ,  τ̂(T_{+}) = Y \frac{∂}{∂X} ,  τ̂(T_{0}) = X \frac{∂}{∂X} + Y \frac{∂}{∂Y} ,
where T_{−} = (\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix}) ,  T_{+} = (\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}) ,  T_{0} = (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix})

are the standard generators in 𝔰𝔩 (2, ℝ). The bases

v_{j} = \frac{4!}{(2 + j)!}σ_{j}, resp., v̂_{j} = \frac{4!}{(2 + j)!}σ̂_{j}, for j = −2, …, 2

are then the canonical ones, i.e.,

τ(T_{0})v_{j} = 2jv_{j}, τ(T_{−})v_{j} = v_{j−1}, τ(T_{+})v_{j} = (2 − j)(4 + j)v_{j+1},

and analogously for τ̂. Now we consider the SL(2, ℝ) orbits of P_{5}. Therefore, let P = ∑ α_{j}σ̂_{j}.

To P we assign the polynomial P̃ = ∑_{j=0}^{4} α_{2−j}ξ^{j}. The group SL(2, ℝ) acts at the roots of the latter polynomial by Möbius transformations of the upper half-plane. In dependence of the distribution of the roots of P̃ we distinguish the following cases:

1. P̃ has at least two different real roots. By means of a Möbius transformation, we send the root of maximal multiplicity to infinity and another root to the origin. One can choose the latter root so that the images of the remaining roots have non-positive real part. Thus P̃ is in the orbit of a polynomial of degree ≤ 3 and without constant term. The remaining freedom is then a map (X, Y) ↦ (μX, μ^{−1}Y) with μ ∈ ℝ^{+}. One achieves that, up to a real multiplier, P takes one of the forms

(a) P = σ̂_{1} + aσ̂_{0} + σ̂_{−1} with a > 2, if all roots are different and real (b) P = σ̂_{1} + aσ̂_{0} + σ̂_{−1} with 0 ≤ a < 2, if there is a pair of complex

conjugate roots in addition to two real ones.

(c) P = σ̂_{1}, if there is a root of multiplicity 3.

(d) P = σ̂_{2} − σ̂_{0}, if there is a root of multiplicity 2 (to be sent to infinity) and two more different real roots (to be sent to ±1).

(e) P = σ̂_{0} in the case of two roots of multiplicity 2.

2. P̃ has only one real root of multiplicity 4. One can send this root to infinity. Thus, this case is equivalent to P = σ̂_{2}.

3. P̃ has a real root of multiplicity 2 and a pair of complex conjugate roots. It is then possible to send the real root to infinity and one complex root to the imaginary unit. This gives

P = σ̂_{2} + σ̂_{0}.

4. P̃ has two pairs of complex conjugate roots. We can achieve that the roots from the upper half-plane shift to the imaginary unity and to a purely imaginary number in the upper half-plane. We obtain

P = σ̂_{2} + aσ̂_{0} + σ̂_{−2}, with a ≥ 2.

The parameter a = 2 corresponds to the case of a double complex root.

By means of the scaling (z, w) ↦ (λz, λ^{2}w) we make the said multiplier equal to ±1. Since the remaining freedom in the change of coordinates (z, w) ↦ (Jz, −w) transforms σ_{j} to −σ_{−j} we can always make the multiplier equal to 1.

We turn now to the pseudo-convex case. Any harmonic polynomial F_{22} can be written in the form

h_{22} = α_{2}η_{2} + α_{1}η_{1} + α_{0}η_{0} + ᾱ_{1}η_{−1} + ᾱ_{2}η_{−2},

with α_{2}, α_{1} ∈ ℂ and c ∈ ℝ. By Q_{5} ⊂ ℂ^{(4)}[X : Y] we denote the subspace of the space of homogeneous polynomials with degree 4 in two variables Q(X, Y) such that Q(Y, X) = \overline{Q(X̄, Ȳ)} with the standard SU(2)-action. The latter action is isomorphic to the SU(2)-action on the space of harmonic polynomials via

η_{2} ↦ X^{4}, η_{1} ↦ X^{3}Y, η_{0} ↦ X^{2}Y^{2}, η_{−1} ↦ XY^{3}, η_{−2} ↦ Y^{4}.

We consider the SU(2) orbits of Q_{5}. Let

Q = α_{2}X^{4} + α_{1}X^{3}Y + α_{0}X^{2}Y^{2} + ᾱ_{1}XY^{3} + ᾱ_{2}Y^{4},
Q̃ = α_{2}ξ^{4} + α_{1}ξ^{3} + α_{0}ξ^{2} + ᾱ_{1}ξ + ᾱ_{2}.

One easily checks that if a root a of Q̃ is not contained in the unit circle then ā^{−1} is also a root. The group SU(2) acts at the roots of the polynomial Q̃ by Möbius transformations of the unit disk. The possible distributions of these roots lead to the following list of the representatives of orbits (up to a real multiplier):

1. Q̃ has at least two different roots and all are unimodular. We send the root of maximal multiplicity to 1 and another root to −1. One can choose the latter root so that the images of the remaining roots are in the upper half-plane. Moreover, if the remaining roots are different, one can achieve that the product of their images is −1 (they are symmetric with respect to the imaginary axis). We obtain the following options

(a) Q = σ̂_{−2} + iaσ̂_{−1} − 2σ̂_{0} − iaσ̂_{1} + σ̂_{2} with 0 ≤ a < 1, if all roots are different and unimodular.

(b) Q = σ̂_{−2} − 2σ̂_{−1} + 2σ̂_{0} − 2σ̂_{1} + σ̂_{2} if there are three unimodular roots, one of them of multiplicity 2. Then the double root will be sent to 1 and the other two to ±i.

(c) Q = iσ̂_{−2} + 2iσ̂_{−1} − 2iσ̂_{1} − iσ̂_{2}, if there is a root of multiplicity 3. The triple root is sent to −1 and the other root to 1.

(d) Q = σ̂_{−2} − 2σ̂_{0} + σ̂_{2}, if there are two double roots (to be sent to ±1). This is (1.a) with a = 1.

2. Q̃ has only one unimodular root of multiplicity 4. One can send this root to −1. Thus, this case is equivalent to Q = σ̂_{−2} + 4σ̂_{−1} + 6σ̂_{0} + 4σ̂_{1} + σ̂_{2}.

3. Q̃ has a two unimodular and a pair of non-unimodular roots. It is then possible to send a non-unimodular root to the origin and the unimodular roots to mutually complex conjugate elements with nonpositive real part. This gives

Q = σ̂_{−1} + aσ̂_{0} + σ̂_{1}, 0 ≤ a ≤ 2,

where a = 2 corresponds to the case of a unimodular double root.

4. Q̃ has two pairs of non-unimodular roots. We can achieve that one root goes to the origin and the other two are real non-positive. We obtain

Q = σ̂_{−1} + aσ̂_{0} + σ̂_{1}, with a > 2.

5. Q̃ has a pair of non-unimodular double roots. Then

Q = σ̂_{0}.

By the scaling (z, w) ↦ (λz, λ^{2}w) we make the multiplier equal to ±1. □

References

[CM74] S.S. Chern and J.Moser. Real hypersurfaces in complex manifolds. Acta Math., 133(3-4):219–271, 1974.