j. differential geometry

63 (2003) 155-170

HOLOMORPHIC DE RHAM COHOMOLOGY OF STRONGLY PSEUDOCONVEX CR MANIFOLDS

WITH S^{1}-ACTIONS

HING SUN LUK & STEPHEN S.-T. YAU

This paper is dedicated to Professor Yum Tong Siu on the occasion of his 60th birthday.

Abstract

In this paper, we study the holomorphic de Rham cohomology of a compact strongly pseudoconvex CR manifold X in ℂ^{N} with a transversal holomorphic S^{1}-action. The holomorphic de Rham cohomology is derived from the Kohn-Rossi cohomology and is particularly interesting when X is of real dimension three and the Kohn-Rossi cohomology is infinite dimensional. In Theorem A, we relate the holomorphic de Rham cohomology H_{h}^{k}(X) to the punctured local holomorphic de Rham cohomology at the singularity in the variety V which X bounds. In case X is of real codimension three in ℂ^{n+1}, we prove that H_{h}^{n−1}(X) and H_{h}^{n}(X) have the same dimension while all other H_{h}^{k}(X), k > 0, vanish (Theorem B). If X is three-dimensional and V has at most rational singularities, we prove that H_{h}^{1}(X) and H_{h}^{2}(X) vanish (Theorem C). In case X is three-dimensional and N = 3, we obtain in Theorem D a complete characterization of the vanishing of the holomorphic de Rham cohomology of X.

1. Introduction

Let M be a complex manifold. The k-th holomorphic de Rham cohomology H_{h}^{k}(M) of M is defined to be the d-closed holomorphic k-forms quotient by the d-exact holomorphic k-forms. It is well-known that if M is a Stein manifold, then the holomorphic de Rham cohomology coincides with the ordinary de Rham cohomology. For any CR manifold

Research partially supported by NSF. Research partially supported by RGC Hong Kong. Received 01/11/2002.

X, one can define the holomorphic de Rham cohomology group H_{h}^{k}(X). One may regard the holomorphic de Rham cohomology as derived from the Kohn-Rossi cohomology, in the sense, following Tanaka [9], that the Kohn-Rossi cohomology groups are the E_{1}^{p,q}(X) terms and the holomorphic de Rham cohomology groups are the E_{2}^{k,0}(X) terms of a natural spectral sequence E_{r}^{p,q}(X) on X.

Let X be a compact connected strongly pseudoconvex CR manifold which admits a transversal holomorphic S^{1}-action φ_{t}. Suppose that X is of dimension 2n−1 (n ⩾ 2) and is CR embeddable in complex Euclidean space. Then it is known [5] that X bounds a compact complex analytic variety V and φ_{t} extends to a weakly holomorphic △^{∗}-action Φ_{t} on V with a single fixed point x where △^{∗} = {z ∈ ℂ : 0 < |z| ⩽ 1}. Further, V has at most a singularity at x. Thus a natural problem is to study how the holomorphic de Rham cohomology of X is related to invariants at the point x.

In [3], we introduce the punctured local holomorphic de Rham cohomology H_{h}^{k}(V, x) at any point x in a complex analytic space V with only isolated singularities. The punctured local holomorphic de Rham cohomology vanishes at regular points. It turns out that the punctured local holomorphic de Rham cohomology is an important local invariant which can be used to tell when an isolated singularity is quasi-homogeneous. Our first main result in this paper is the following theorem.

Theorem A. Let X be a compact connected (2n − 1)-dimensional (n ⩾ 2) strongly pseudoconvex CR manifold with a transversal holomorphic S^{1}-action. Suppose that X bounds a complex analytic variety V in ℂ^{N} with x as the only singular point of V. Then the holomorphic de Rham cohomology of X is naturally isomorphic to the punctured local holomorphic de Rham cohomology of V at x. That is, H_{h}^{k}(X) ≅ H_{h}^{k}(V, x) for all k ⩾ 0.

Theorem A is proved by a homotopy argument, which works also when the fixed point x of the extended △^{∗}-action Φ_{t} is a regular point of V, showing that H_{h}^{k}(X) = 0 for all k ⩾ 1 in such cases. By Theorem A and results in [3] on punctured local holomorphic de Rham cohomology, we get

Theorem B. Let X be a compact connected (2n − 1)-dimensional (n ⩾ 2) strongly pseudoconvex CR manifold with a transversal holomorphic S^{1}-action. Suppose that X bounds a compact complex analytic hypersurface V in ℂ^{n+1}. Then:

(1) dim H_{h}^{k}(X) = 0 1 ⩽ k ⩽ n − 2.

(2) dim H_{h}^{n−1}(X) = dim H_{h}^{n}(X).

The holomorphic de Rham cohomology is particularly interesting when X is three dimensional, in which case the Kohn-Rossi cohomology is infinite dimensional. The simplest singularities that a three dimensional X may bound in ℂ^{3} are the rational singularities of types A_{n}, D_{n}, E_{6}, E_{7}, E_{8} and we prove the following theorem.

Theorem C. Let X be a compact connected three dimensional strongly pseudoconvex CR manifold with transversal holomorphic S^{1}- action. Suppose that X bounds a compact complex analytic variety V in ℂ^{N} with at most rational singularities. Then H_{h}^{1}(X) = 0 and H_{h}^{2}(X) = 0.

We would like to remark that in the statement of Theorem C, X is not required to be in ℂ^{3} and the singularity in V may not be of types A_{n}, D_{n}, E_{6}, E_{7} and E_{8}. If X is in ℂ^{3}, then the following theorem gives a complete characterization of when the holomorphic de Rham cohomology groups of X vanish.

Theorem D. Let X in ℂ^{3} be a compact connected 3-dimensional strongly pseudoconvex CR manifold with transversal holomorphic S^{1}- action. Then X bounds a compact complex analytic variety V in ℂ^{3} with at most one singularity. Let M be a resolution of V with A as exceptional set. Then H_{h}^{1}(X) = 0 and H_{h}^{2}(X) = 0 if and only if H^{1}(A, ℂ) = 0.

Our assumption of the existence of a transversal holomorphic S^{1}- action on the CR manifold is a natural assumption arising from the work of Lawson and Yau [5]. Such CR manifolds include the intersection of any quasihomogeneous hypersurface in ℂ^{n+1} with a sphere centered at the origin.

In Section 2, we recall the intrinsic definition of the holomorphic de Rham cohomology of CR manifolds and clarify the extrinsic formulation in the boundary case (Theorem 2.7). Theorems A, B, C and D are proved in Section 3. We prove Theorem C using a proposition of Campana and Flenner [2].

Part of this work was done while the second author was a Ze Jiang Visiting Professor at East China Normal University.

2. Holomorphic de Rham cohomology of CR manifolds

Kohn-Rossi cohomology was first introduced by Kohn-Rossi [4]. Following Tanaka [9], we reformulate the definition in a way independent of the interior manifold and derive the holomorphic de Rham cohomology from the Kohn-Rossi cohomology.

Definition 2.1. Let X be a connected orientable manifold of real dimension 2n−1, n ⩾ 2. A CR structure on X is a rank n−1 subbundle S of the complexified tangent bundle ℂT (X) such that:

(1) S ∩ \overline{S} = \{0\}.

(2) If L, L′ are local sections of S, then so is [L, L′].

A manifold with a CR structure is called a CR manifold. Then there is a unique subbundle H of T (X) and a unique homomorphism J : H ⟶ H such that ℂH = S⊕\overline{S}, J^{2} = −1 and S = \{v−iJv : v ∈ H\}.

For any C^{∞} function u, there is a section \overline{∂}_{b}u ∈ Γ(\overline{S}^{∗}) defined by (\overline{∂}_{b}u)(\overline{L}) = \overline{L}u for any L ∈ Γ(S). This can be generalized as follows:

Definition 2.2. A complex vector bundle E over X is said to be holomorphic if there is a differential operator \overline{∂}_{E} : Γ(E) ⟶ Γ(E ⊗ \overline{S}^{∗}) such that if \overline{L}u denotes (\overline{∂}_{E}u)(\overline{L}) for u ∈ Γ(E) and L ∈ Γ(S), then for any L_{1}, L_{2} ∈ Γ(S) and any C^{∞} function f on X:

(1) \overline{L}(fu) = (\overline{L}f)u + f(\overline{L}u).

(2) [\overline{L}_{1}, \overline{L}_{2}]u = \overline{L}_{1}\overline{L}_{2}u − \overline{L}_{2}\overline{L}_{1}u.

A solution u of the equation \overline{∂}_{E}u = 0 is called a holomorphic section.

The vector bundle T̂(X) = ℂT (X)/\overline{S} is holomorphic with respect to the following \overline{∂} = \overline{∂}_{T̂(X)}. Let ω be the projection from ℂT (X) to T̂(X). Take any u ∈ Γ(T̂(X)) and express it as u = ω(Z), Z ∈ Γ(ℂT (X)). For any L ∈ Γ(S), define (\overline{∂}u)(\overline{L}) = ω([\overline{L}, Z]). The section (\overline{∂}u)(\overline{L}) of T̂(X) does not depend on the choice of Z and \overline{∂}u gives a section of T̂(X)⊗\overline{S}^{∗}. Further the operator \overline{∂} satisfies the conditions in Definition 2.2. The resulting holomorphic vector bundle T̂(X) is called the holomorphic tangent bundle of X.

Lemma 2.3. If X is a real hypersurface in a complex manifold M, then the holomorphic tangent bundle T̂(X) is naturally isomorphic to the restriction to X of the bundle T^{1,0}(M) of all (1, 0) tangent vectors to M.

Proof. The CR structure of X is given by S = ℂT (X) ∩ T^{1,0}(M). The inclusion ℂT (X) ⟶ ℂT (M) induces a vector bundle isomorphism ℂT (X)/\overline{S} ⟶ ℂT (M)/T^{0,1}(M), which we write as Φ : T̂(X) ⟶ T^{1,0}(M)|_{X}. T^{1,0}(M)|_{X} is a holomorphic vector bundle with respect to the following natural \overline{∂} operator. For L ∈ Γ(S) and Y ∈ Γ(T^{1,0}(M)|_{X}), (\overline{∂}Y)(\overline{L}) = \overline{L}(Y) is defined to be the (1, 0) component of [\overline{L}, Y]. If

Y = ∑ y^{i} \frac{∂}{∂z^{j}} in local coordinates of M, then \overline{L}(Y) = ∑(\overline{L}y^{j})\frac{∂}{∂z^{j}}. It follows that Φ is an isomorphism of holomorphic vector bundles in the obvious sense that \overline{L}Φu = Φ\overline{L}u for any u ∈ Γ(T̂(X)). q.e.d.

For a holomorphic vector bundle E over X, set

C^{q}(X, E) = E ⊗ ^{⋀q} \overline{S}^{∗}, C^{q}(X, E) = Γ(C^{q}(X, E)).

\overline{∂}_{E} induces a differential operator \overline{∂}_{E}^{q} : C^{q}(X, E) ⟶ C^{q+1}(X, E) as follows. For any ψ ∈ C^{q}(X, E), (\overline{∂}_{E}^{q}ψ)(\overline{L}_{1}, …, \overline{L}_{q+1}) = ∑_{i}(−1)^{i+1} \overline{L}_{i}(ψ(\overline{L}_{1}, …, ˆ \overline{L}_{i}, …, \overline{L}_{q+1})) + ∑_{i<j}(−1)^{i+j} ψ([\overline{L}_{i}, \overline{L}_{j}], \overline{L}_{1}, …, ˆ \overline{L}_{i}, …,

ˆ\overline{L}_{j}, …, \overline{L}_{q+1}), for all L_{1}, …, L_{q+1} ∈ Γ(S), defines an element \overline{∂}_{E}^{q}ψ ∈ C^{q+1}(X, E). Further \overline{∂}_{E}^{q+1}\overline{∂}_{E}^{q} = 0. The cohomology groups of the resulting complex \{C^{q}(X, E), \overline{∂}_{E}^{q}\} are denoted by H^{q}(X, E).

The CR structure S on X induces a natural filtration of the de Rham complex {A^{k}(X), d} with complex coefficients: Let A^{k}(X) = ⋀^{k} ℂT (X)^{∗} and denote by F^{p}(A^{k}(X)) the subbundle of A^{k}(X) consisting of all ϕ satisfying ϕ(Y_{1}, …, Y_{p−1}, \overline{Z}_{1}, …, \overline{Z}_{k−p+1}) = 0 for all Y ’s in ℂT (X) and Z’s in S. Then A^{k}(X) = F^{0}(A^{k}(X)) ⊃ F^{1}(A^{k}(X)) ⊃ ⋯ ⊃ F^{k}(A^{k}(X)) ⊃ F^{k+1}(A^{k}(X)) = 0. Set F^{p}(A^{k}(X)) = Γ(F^{p}(A^{k}(X)). Since dF^{p}(A^{k}(X)) ⊂ F^{p}(A^{k+1}(X)), the collection {F^{p}(A^{k}(X))} gives a filtration of the de Rham complex.

Consider the spectral sequence {E_{r}^{p,q}(X), d_{r}} associated with the filtration {F^{p}(A(X))}. Let

A^{p,q}(X) = F^{p}(A^{p+q}(X)),  A^{p,q}(X) = Γ(A^{p,q}(X)),
C^{p,q}(X) = A^{p,q}(X)/A^{p+1,q−1}(X),  C^{p,q}(X) = Γ(C^{p,q}(X)).

First, E_{0}^{p,q}(X) = C^{p,q}(X) and d_{0} : C^{p,q}(X) ⟶ C^{p,q+1}(X) is the map induced by d. Note that E_{0}^{k,0}(X) = C^{k,0}(X) = A^{k,0}(X) ⊂ A^{k}(X).

Next, E_{1}^{p,q}(X) = \frac{Ker(d_{0} : C^{p,q}(X) ⟶ C^{p,q+1}(X))}{Im(d_{0} : C^{p,q−1}(X) ⟶ C^{p,q}(X))} and d_{1} : E_{1}^{p,q}(X)

⟶ E_{1}^{p+1,q}(X) is the naturally induced map.

In particular,

E_{1}^{k,0}(X) = Ker(d_{0} : C^{k,0}(X) ⟶ C^{k,1}(X))
= {φ ∈ A^{k,0}(X) : dφ ∈ A^{k+1,0}(X)}

and d_{1} is just d on E_{1}^{k,0}(X) ⊂ A^{k}(X).

E_{1}^{k,0}(X) is called the space of holomorphic k-forms on X. Denoting E_{1}^{k,0}(X) by S^{k}(X), we have the holomorphic de Rham complex {S^{k}(X), d}.

Then,

E_{2}^{k,0}(X) = \frac{Ker(d : S^{k}(X) ⟶ S^{k+1}(X))}{Im(d : S^{k−1}(X) ⟶ S^{k}(X))}
= \frac{\{closed holomorphic k-forms on X\}}{\{exact holomorphic k-forms on X\}}.

The groups E_{1}^{p,q}(X) and E_{2}^{k,0}(X) will be denoted by H_{KR}^{p,q}(X) and H_{h}^{k}(X) respectively. The former are the Kohn-Rossi cohomology groups and the latter are the holomorphic de Rham cohomology groups.

For computations, we need a more explicit description of H_{KR}^{p,q}(X) and H_{h}^{k}(X) in case X is a real hypersurface in a complex manifold M. In this case, the proof of Lemma 2.3 provides an isomorphism Φ : T̂(X) ⟶ T^{1,0}(M)|_{X} of holomorphic vector bundles over X. For any holomorphic vector bundle E over X, ⋀^{p} E^{∗} is naturally a holomorphic vector bundle over X with respect to the following \overline{∂} : Γ(⋀^{p} E^{∗}) ⟶ Γ(⋀^{p} E^{∗} ⊗ \overline{S}^{∗}). Let θ ∈ Γ(⋀^{p} E^{∗}) and L ∈ Γ(S). (\overline{∂}θ)(\overline{L}) = \overline{L}θ is defined by

(\overline{L}θ)(u_{1}, …, u_{p}) = \overline{L}(θ(u_{1}, …, u_{p})) − ∑_{i=1}^{p} θ(u_{1}, …, \overline{L}u_{i}, …, u_{p})

for all u_{1}, …, u_{p} ∈ Γ(E). Let both ⋀^{p} T̂(X)^{∗} and ⋀^{p} T^{1,0}(M)|_{X}^{∗} be made holomorphic vector bundles over X in this way. Then Φ induces an isomorphism Φ^{∗} : ⋀^{p} T^{1,0}(M)|_{X}^{∗} ⟶ ⋀^{p} T̂(X)^{∗} of holomorphic vector bundles over X, where (Φ^{∗}ξ)(u_{1}, …, u_{p}) = ξ(Φu_{1}, …, Φu_{p}) for ξ ∈ ⋀^{p} T^{1,0}(M)|_{X}^{∗}, u_{1}, …, u_{p} ∈ T̂(X). It is useful to write down the \overline{∂} operator of ⋀^{p} T^{1,0}(M)|_{X}^{∗} in local holomorphic coordinates z_{1}, …, z_{n} of M. Any ξ ∈ Γ(⋀^{p} T^{1,0}(M)|_{X}^{∗}) can be written as ξ = ∑ ξ_{i_{1}...i_{p}}dz^{i_{1}} ∧⋯∧ dz^{i_{p}}, ξ_{i_{1}...i_{p}} being local functions on X. For any L ∈ Γ(S), one checks that \overline{L}ξ = ∑(\overline{L}ξ_{i_{1}...i_{p}})dz^{i_{1}} ∧ ⋯ ∧ dz^{i_{p}}.

Lemma 2.4. Let X be a real hypersurface in a complex manifold M.

(1) The two complexes { C^{p,q} (X), d_{0} } and { C^{q} (X, ⋀^{p} T^{1,0}M|_{X}^{∗}), (−1)^{p}\overline{∂}^{q}\} can be naturally identified.

(2) S^{k}(X) can be identified with the subspace of Γ(⋀^{k} T^{1,0}M|_{X}^{∗}) consisting of those ξ = ∑ ξ_{I}dz^{I} satisfying \overline{∂}_{b}ξ_{I} = 0 for all I.

Proof. First consider the map ι^{p} : A^{p,q}(X) ⟶ C^{q}(X, ⋀^{p} T̂(X)^{∗}) = ⋀^{p} T̂ (X)^{∗} ⊗ ⋀^{q} \overline{S}^{∗} defined by (ι^{p}ϕ) (ω(Z_{1}), …, ω(Z_{p}); \overline{L}_{1}, …, \overline{L}_{q}) = ϕ(Z_{1}, …, Z_{p}, \overline{L}_{1}, …, \overline{L}_{q}) for all ϕ ∈ A^{p,q}(X), ω(Z_{i}) ∈ T̂(X) where Z_{i} ∈ ℂT (X), and L_{j} ∈ S. ι^{p} is surjective, with kernel A^{p+1,q−1}(X). Hence ι^{p} induces an isomorphism C^{p,q}(X) ⟶ C^{q}(X, ⋀^{p} T̂(X)^{∗}) which we also denote by ι^{p}. A careful checking shows that for any φ ∈ A^{p,q}(X), ι^{p}dφ = (−1)^{p}\overline{∂}^{q}ι^{p}φ. Thus \{C^{p,q}(X), d_{0}\} and \{C^{q}(X, ⋀^{p} T̂(X)^{∗}), (−1)^{p}\overline{∂}^{q}\} can be naturally identified. Since ⋀^{p} T̂(X)^{∗} and ⋀^{p} T^{1,0}(M)|_{X}^{∗} are isomorphic as holomorphic vector bundles over X, (1) follows. By (1), S^{k}(X) = Ker(d_{0} : C^{k,0}(X) ⟶ C^{k,1}(X)) can be identified with Ker(\overline{∂} : C^{0}(X, ⋀^{k} T^{1,0}(M)|_{X}^{∗}) ⟶ C^{1}(X, ⋀^{k} T^{1,0} (M)|_{X}^{∗})). Then (2) follows from the remark preceeding the lemma. q.e.d.

Definition 2.5. Let L_{1}, …, L_{n−1} be a local frame of the CR structure S on X, so that \overline{L}_{1}, …, \overline{L}_{n−1} is a local frame of \overline{S}. Choose a local section N of ℂT (X) such that L_{1}, …, L_{n−1}, \overline{L}_{1}, …, \overline{L}_{n−1}, N span ℂT (X). Assuming that N is purely imaginary,

[L_{i}, \overline{L}_{j}] = ∑ a_{ij}^{k}L_{k} + ∑ b_{ij}^{k}\overline{L}_{k} + c_{ij}N

where (c_{ij}) is a Hermitian matrix called the Levi form of X. X is said to be strongly pseudoconvex if the Levi form is definite at each point of X. This condition is independent of the choice of L_{1}, …, L_{n−1}, N.

Lemma 2.6. Let X be a compact connected strongly pseudoconvex real hypersurface in a complex manifold M. If all φ ∈ S^{k}(X) can be extended to holomorphic k-forms on a common one-sided neighborhood U of X in M, then the complex {S^{k}(X), d} can be naturally identified with the complex {Γ(U, Ω^{k}), d}, where Γ(U, Ω^{k}) denotes the space of holomorphic k-forms on U \ X which extend smoothly (C^{∞}) up to the boundary component X.

Proof. We have seen that any φ ∈ S^{k}(X) can be identified with some ξ ∈ Γ(⋀^{k} T^{1,0}(M)|_{X}^{∗}) such that φ(W_{1}, …, W_{k}) = ξ(W_{1}^{1,0}, …, W_{k}^{1,0}) where W_{i} ∈ Γ(ℂT X) and W_{i}^{1,0} denotes the (1,0) component of W_{i} in ℂT M. Denote the holomorphic extension of ξ to U also by ξ. One defines dξ(W_{1}, …, W_{k+1}), W_{i} ∈ Γ(ℂT X), in the obvious way and checks that dφ(W_{1}, …, W_{k+1}) = dξ(W_{1}, …, W_{k+1}), in particular when W_{k+1} = N, N as in Definition 2.5. q.e.d.

Finally, we have the following situation.

Theorem 2.7. Let X be a compact connected (2n − 1)-dimensional (n ⩾ 2) strongly pseudoconvex CR manifold. Suppose that X is the boundary of a compact n-dimensional strongly pseudoconvex manifold M which is a modification of a compact Stein space V with only isolated singularities. Let A be the minimal compact analytic set in M which can be blown down to the isolated singularities. Then the complex {S^{k}(X), d} can be naturally identified with the complex {Γ(M\A, Ω^{k}), d} (notation as in Lemma 2.6).

Proof. One can take a one-convex exhaustion function ρ on the interior of M such that ρ ⩾ 0 everywhere and ρ(y) = 0 if and only if y ∈ A. Any φ ∈ S^{k}(X) can be extended to a one-sided neighborhood U of X in M, where U may be taken to be M \ {ρ ⩽ r}. By Andreotti and Grauert [1, Théorème 15], Γ(M \ {ρ ⩽ r}, Ω_{M\{ρ⩽r}}^{k}) is isomorphic to Γ(M \ A, Ω_{M\A}^{k}). The rest is as in Lemma 2.6. q.e.d.

3. Transversal holomorphic S^{1}-action and punctured holomorphic de Rham cohomology

In the following, we consider a compact connected strongly pseudoconvex CR manifold in ℂ^{N} which admits a transversal holomorphic S^{1}-action. We recall:

Definition 3.1. Let X be a CR manifold as in Definition 2.1. A smooth S^{1}-action (of class C^{1}) on X is said to be holomorphic if it preserves the subspaces H ⊂ T (X) and commutes with J. It is said to be transversal if, in addition, the vector field v which generates the action, is transversal to H at all points of X.

Definition 3.2. Let φ_{t} be a continuous S^{1}-action on a complex analytic space V which preserves the regular points of V. The action is called weakly holomorphic (resp. holomorphic) if for all t, φ_{t}^{∗}O_{V}^{ω} = O_{V}^{ω} (resp. φ_{t}^{∗}O_{V} = O_{V}), where O_{V}^{ω} is the sheaf of germs of weakly holomorphic functions on V (i.e., the sheaf of germs of locally bounded holomorphic functions on the regular part of V).

Theorem 3.3 ([5]). Let X be a compact connected CR manifold which bounds a compact complex analytic variety V of dimension n in ℂ^{N}. Then any transversal holomorphic S^{1}-action on X extends to a weakly holomorphic representation Φ of the analytic semigroup △^{∗} = {t ∈ ℂ : 0 < |t| ⩽ 1} as a semigroup of analytic embeddings of V into itself. This action has a single fixed point x and given any neighborhood U of x in V, there is an ε > 0 so that Φ_{t}(V) ⊂ U for all |t| < ε.

Indeed V has at most one singularity, namely at x. The following theorem and its proof is our key to compute the holomorphic de Rham cohomology H_{h}^{k}(X).

Theorem 3.4. Let X be a compact connected (2n − 1)-dimensional (n ⩾ 2) strongly pseudoconvex CR manifold with transversal holomorphic S^{1}-action φ_{t}. Suppose that X bounds a compact complex analytic variety V in ℂ^{N} with x as its only singular point. Let U be any open neighborhood of x contained in the interior V^{∘} of V such that under the extended weakly holomorphic △^{∗}-action, Φ_{t}(U) ⊂ U for all t ∈ △^{∗}. Then the holomorphic de Rham cohomology groups of the complex manifolds V^{∘} \ {x} and U \ {x} are naturally isomorphic. More precisely, the

restriction map r : H_{h}^{k}(V^{∘} \ {x}) ⟶ H_{h}^{k}(U \ {x}) is an isomorphism, k ⩾ 0.

Proof. By Theorem 3.3, φ_{t} extends to a weakly holomorphic △^{∗}- action Φ_{t} and there exists t_{0} ∈ △^{∗} such that V^{∘} ⊃ U ⊃ Φ_{t_{0}}(V^{∘}) ⊃ Φ_{t_{0}}(U). Consider the following natural maps induced by restrictions:

H_{h}^{k}(V^{∘} \ {x}) ⟶^{r} H_{h}^{k}(U \ {x}) ⟶^{r_{1}}

H_{h}^{k}(Φ_{t_{0}}(V^{∘}) \ {x}) ⟶^{r_{2}} H_{h}^{k}(Φ_{t_{0}}(U) \ {x}).

We first prove that the composition

H_{h}^{k}(V^{∘} \ {x}) ⟶^{r_{1}∘r} H_{h}^{k}(Φ_{t_{0}}(V^{∘}) \ {x})

is equal to the map H_{h}^{k}(V^{∘} \ {x}) ⟶^{Φ_{t_{0}}^{−1∗}} H_{h}^{k}(Φ_{t_{0}}(V^{∘}) \ {x}) induced by the inverse of the analytic embedding Φ_{t_{0}}. This can be done by means of a chain homotopy as follows. Let △_{t_{0}}^{∗} = {t ∈ ℂ : |t_{0}| ⩽ |t| ⩽ 1}. For

164 h.s. luk & s. s.-t. yau

each t ∈ △_{t_{0}}^{∗}, we have V^{∘} \ \{x\} \overrightarrow{⟵}_{Φ_{t}^{−1}}^{Φ_{t}} Φ_{t}(V^{∘}) \ \{x\} ⊃ Φ_{t_{0}}(V^{∘}) \ \{x\}.

Write Ψ_{t} = Φ_{t}^{−1}|_{Φ_{t_{0}}(V^{∘})\{x}} and factorize it by

Φ_{t_{0}}(V^{∘}) \ {x} ⟶^{λ_{t}} (Φ_{t_{0}}(V^{∘}) \ {x}) × △_{t_{0}}^{∗} ⟶^{Ψ} V^{∘} \ {x}
z ⟼ (z, t) ⟼ Ψ_{t}(z).

Thus Ψ_{t} = Ψ ∘ λ_{t}, where Ψ is holomorphic in z and t. An arbitrary holomorphic k-form ω^{∗} on (Φ_{t_{0}}(V^{∘}) \ {x}) × △_{t_{0}}^{∗} may be written in local holomorphic coordinates z_{1}, …, z_{n} of Φ_{t_{0}}(V^{∘}) \ {x} as ω^{∗} = ∑ f_{I}(z, t)dz_{I} + ∑ g_{J}(z, t)dt ∧ dz_{J}. (3.1) We have forms λ_{t}^{∗}ω^{∗} = ∑ f_{I}(z, t)dz_{I} on Φ_{t_{0}}(V^{∘})\{x}. Then on (Φ_{t_{0}}(V^{∘})\ {x}) × △_{t_{0}}^{∗},

\frac{d}{dt}λ_{t}^{∗}ω^{∗} = ∑ \frac{∂f_{I}}{∂t} (z, t)dz_{I}. (3.2)

Also,

(3.3)  dω^{∗} = ∑ \frac{∂f_{I}}{∂z_{i}} (z, t)dz_{i} ∧ dz_{I} + ∑ \frac{∂f_{I}}{∂t} (z, t)dt ∧ dz_{I}

+ ∑ \frac{∂g_{J}}{∂z_{i}} (z, t)dz_{i} ∧ dt ∧ dz_{J}

i (\frac{∂}{∂t}) dω^{∗} = ∑ \frac{∂f_{I}}{∂t} (z, t)dz_{I} − ∑ \frac{∂g_{J}}{∂z_{i}} (z, t)dz_{i} ∧ dz_{J}
λ_{t}^{∗} (i (\frac{∂}{∂t}) dω^{∗}) = ∑ \frac{∂f_{I}}{∂t} (z, t)dz_{I} − ∑ \frac{∂g_{J}}{∂z_{i}} (z, t)dz_{i} ∧ dz_{J}.

On the other hand,

(3.4)  i (\frac{∂}{∂t}) ω^{∗} = ∑ g_{J}(z, t)dz_{J}
d (i (\frac{∂}{∂t}) ω^{∗}) = ∑ \frac{∂g_{J}}{∂z_{i}} (z, t)dz_{i} ∧ dz_{J} + ∑ \frac{∂g_{J}}{∂t} (z, t)dt ∧ dz_{J}
λ_{t}^{∗} (d (i (\frac{∂}{∂t}) ω^{∗})) = ∑ \frac{∂g_{J}}{∂z_{i}} (z, t)dz_{i} ∧ dz_{J}.

Treating (3.3) and (3.4) like (3.2) as equations on (Φ_{t_{0}}(V^{∘}) \ {x}) × △_{t_{0}}^{∗}, we get

\frac{d}{dt}λ_{t}^{∗}ω^{∗} = λ_{t}^{∗} (i (\frac{∂}{∂t}) dω^{∗}) + d (λ_{t}^{∗} (i (\frac{∂}{∂t}) ω^{∗})) . (3.5)

For any holomorphic k-form ω on V^{∘} \ {x}, putting ω^{∗} = Ψ^{∗}ω in (3.5) gives

\frac{d}{dt}Ψ_{t}^{∗}ω = (λ_{t}^{∗}i (\frac{∂}{∂t}) Ψ^{∗}) dω + d (λ_{t}^{∗}i (\frac{∂}{∂t}) Ψ^{∗}) ω. (3.6)

Taking a smooth curve γ in △_{t_{0}}^{∗} joining t_{0} to 1 and integrating (3.6) along γ gives

Ψ_{1}^{∗}ω − Ψ_{t_{0}}^{∗}ω = Hdω + dHω (3.7)

where Hη = ∫_{t_{0}}^{1} λ_{t}^{∗}i (\frac{∂}{∂t}) Ψ^{∗}ηdt provides the desired chain homotopy.

Observe that Ψ_{1} is the inclusion map Φ_{t_{0}}(V^{∘}) \ {x} ⟶ V^{∘} \ {x} while

Ψ_{t_{0}} is the biholomorphic map Φ_{t_{0}}^{−1} : Φ_{t_{0}}(V^{∘})\{x} ⟶ V^{∘} \{x}. Hence Ψ_{1}^{∗} (respectively Ψ_{t_{0}}^{∗}) induces r_{1} ∘ r (respectively Φ_{t_{0}}^{−1}^{∗}) on the cohomology level. Then (3.7) implies that r_{1} ∘ r = Φ_{t_{0}}^{−1}^{∗}.

Now, since Φ_{t_{0}}^{−1} is a biholomorphic map, Φ_{t_{0}}^{−1∗} is an isomorphism, hence so is r_{1} ∘ r. On the other hand, if U is a strongly pseudoconvex neighborhood of x in V^{∘} satisfying U ⊃ Φ_{t}(U) for all t ∈ △^{∗}, then we

also have U \ \{x\} \overrightarrow{⟵}_{Φ_{t}^{−1}}^{Φ_{t}} Φ_{t}(U) \ \{x\} ⊃ Φ_{t_{0}}(U) \ \{x\} for all t ∈ △_{t_{0}}^{∗}.

By repeating the above argument, we see that

r_{2} ∘ r_{1}( = Φ_{t_{0}}^{−1}^{∗}) : H_{h}^{k}(U \ {x}) ⟶ H_{h}^{k}(Φ_{t_{0}}(U) \ {x})

is an isomorphism. r_{1} ∘ r and r_{2} ∘ r_{1} bijective implies r_{1} bijective. It follows that r is also bijective, hence an isomorphism. q.e.d.

Remark 3.5. For the simplest case where X is the unit sphere and V is the closed unit ball in ℂ^{2}, the S^{1}-action φ_{t}(z_{1}, z_{2}) = (tz_{1}, tz_{2}) on X extends to the △^{∗}-action Φ_{t}(z_{1}, z_{2}) = (tz_{1}, tz_{2}) on V with fixed point at the origin 0. In this case, if r : ℝ_{+} × ℝ_{+} ⟶ ℝ is C^{∞} and strictly increasing in each variable, and if r(|z_{1}|, |z_{2}|) < 0 defines an open neighborhood U of 0 contained in the open unit ball, then Φ_{t}(U) ⊂ U

for all t ∈ △^{∗}. For example, for 0 < a_{1}, a_{2} < 1, \frac{|z_{1}|^{2}}{a_{1}^{2}} + \frac{|z_{2}|^{2}}{a_{2}^{2}} < 1 defines

such a U which is biholomorphically equivalent to the open unit ball.

On the other hand, there are strongly pseudoconvex neighborhoods U_{1} of 0 in V for which the condition for all t ∈ △^{∗}, Φ_{t}(U_{1}) ⊂ U_{1} fails. For example, let U_{1} be defined by |z_{1}|^{2} + |z_{2}|^{2} + λ(z_{1} + \overline{z}_{1}) < \frac{1}{2} , where λ ∈ ℝ. If 0 < λ < \frac{1}{6}, one checks that:

(1) U_{1} is a strongly pseudoconvex domain contained in the open unit ball.

(2) For (z_{1}, z_{2}) ∈ ∂U_{1}, z_{1} = x_{1} + iy_{1} and t = cos θ + i sin θ, x_{1} < 0 and y_{1} sin θ < 0 implies that (tz_{1}, tz_{2}) ∉ \overline{U_{1}}.

A slight variation of the above proof yields the following corollary.

Corollary 3.6. Let X and (V, x) be as in Theorem 3.4. Denote

by Γ(V^{∘} \ {x}, Ω^{k}) the space of holomorphic k-forms on V^{∘} \ {x} and by Γ(V \{x}, Ω^{k}) the subspace consisting of those holomorphic k-forms that extend smoothly (C^{∞}) up to the boundary component X. Then the two

complexes {Γ(V^{∘} \ {x}, Ω^{•}), d} and {Γ(V \ {x}, Ω^{•}), d} have isomorphic

cohomology groups H_{h}^{k}(V^{∘} \ {x}) and H_{h}^{k}(V \ {x}) respectively.

Proof. Consider the following chain maps

Γ(V \ {x}, Ω^{•}) ⟶^{r} Γ(V^{∘} \ {x}, Ω^{•}) ⟶^{r_{1}}

Γ(Φ_{t_{0}}(V) \ {x}, Ω^{•}) ⟶^{r_{2}} Γ(Φ_{t_{0}}(V^{∘}) \ {x}, Ω^{•})

for any |t_{0}| < 1. Then we have the induced maps

H_{h}^{k}(V \ {x}) ⟶^{r} H_{h}^{k}(V^{∘} \ {x}) ⟶^{r_{1}}

H_{h}^{k}(Φ_{t_{0}}(V) \ {x}) ⟶^{r_{2}} H_{h}^{k}(Φ_{t_{0}}(V^{∘}) \ {x}).

To prove r_{1} ∘ r = Φ_{t_{0}}^{−1}^{∗}, we consider V \ \{x\} \overrightarrow{⟵}_{Φ_{t}^{−1}}^{Φ_{t}} Φ_{t}(V^{∘}) \ \{x\}

⊃ Φ_{t_{0}}(V^{∘}) \ {x} for all t ∈ △_{t_{0}}^{∗}. For any ω ∈ Γ(V \ {x}, Ω^{k}), Ψ_{1}^{∗}ω −

Ψ_{t_{0}}^{∗}ω = Hdω + dHω holds with Hω ∈ Γ(V \ {x}, Ω^{k}). It follows that r_{1} ∘ r = Φ_{t_{0}}^{−1}^{∗}, hence is an isomorphism.

The same argument for V^{∘} in place of V shows that r_{2} ∘ r_{1} is an isomorphism. As before, we conclude that r is an isomorphism. q.e.d.

It is now natural to relate to the theory of punctured local holomorphic de Rham cohomology developed in [3].

Definition 3.7. Let V be a complex analytic space with only isolated singularities. For any x ∈ V, the punctured local holomorphic de Rham cohomology H_{h}^{k}(V, x) is defined to be \underrightarrow{lim} H_{h}^{k}(U \ \{x\}), where U runs over strongly pseudoconvex open neighborhoods of x in V.

We are now ready to prove Theorem A.

Proof of Theorem A. By Theorem 2.7, the complexes {S^{•}(X), d} and {Γ(M \ A, Ω^{•}), d} are isomorphic. The latter is clearly isomorphic to {Γ(V \ {x}, Ω^{•}), d}. Hence H_{h}^{k}(X) is isomorphic to H_{h}^{k}(V \ {x}). By

Corollary 3.6, H_{h}^{k}(X) is isomorphic to H_{h}^{k}(V^{∘} \ {x}) . It suffices to check

that \underrightarrow{lim} H_{h}^{k}(U \ \{x\}) = H_{h}^{k}(V^{∘} \ \{x\}).

We shall define, for any open neighborhood U of x in V^{∘} satisfying the condition in Theorem 3.4, a homomorphism m_{U} : H_{h}^{k}(U \ {x}) ⟶

H_{h}^{k}(V^{∘} \ {x}), such that the following conditions hold:

(1) For any open neighborhoods U ⊃ W of x in V^{∘}, if

r_{W U} : H_{h}^{k}(U \ {x})⟶H_{h}^{k}(W \ {x})

is the map induced by restriction, then m_{W} r_{W U} = m_{U}.

(2) If there are homomorphisms n_{U} : H_{h}^{k}(U \ {x}) ⟶ N into some group N satisfying n_{W} r_{W U} = n_{U} for all open neighborhoods U ⊃ W of x in V^{∘}, then there is a unique homomorphism

n : H_{h}^{k}(V^{∘} \ {x}) ⟶ N

such that n ∘ m_{U} = n_{U}.

For any open neighborhood U of x in V^{∘}, take t_{0} ∈ △^{∗} such that U ⊃ Φ_{t_{0}}(V^{∘}) and set m_{U} = Φ_{t_{0}}^{∗}r_{Φ_{t_{0}}(V^{∘}),U}. To check that m_{U} is independent of the choice of t_{0}, observe that if |t_{1}| ⩽ |t_{0}|, then by the proof of

Theorem 3.4, Φ−1_{\frac{t_{1}}{t_{0}}}^{∗} = r : H_{h}^{k}(Φ_{t_{0}}(V^{∘})\\{x\}) ⟶ H_{h}^{k}(Φ_{t_{1}}(V^{∘})\\{x\}), hence

Φ_{t_{0}}^{∗}r_{Φ_{t_{0}}(V^{∘}),U} = Φ_{t_{1}}^{∗}Φ−1∗_{\frac{t_{1}}{t_{0}}}r_{Φ_{t_{0}}(V^{∘}),U} = Φ_{t_{1}}^{∗}r_{Φ_{t_{1}}(V^{∘}),U}.

To check (1), take t ∈ △^{∗} such that U ⊃ W ⊃ Φ_{t}(V^{∘}). Then m_{W} r_{W U} = Φ_{t}^{∗}r_{Φ_{t}(V^{∘}),W} r_{W U} = m_{U}. To check (2), take n = nV^{∘}. Then nm_{U} =

nV^{∘} Φ_{t}^{∗}r_{Φ_{t}(V^{∘}),U}, where t ∈ △^{∗} and U ⊃ Φ_{t}(V^{∘}). By Theorem 3.4, Φ_{t}^{−1∗} =

r_{Φ_{t}(V^{∘}),V^{∘}}, hence the given condition n_{Φ_{t}(V^{∘})}r_{Φ_{t}(V^{∘}),V^{∘}} = nV^{∘} may be written

as n_{Φ_{t}(V^{∘})} = nV^{∘} Φ_{t}^{∗}. Then nm_{U} = n_{Φ_{t}(V^{∘})}r_{Φ_{t}(V^{∘}),U} = n_{U}. The uniqueness

of n is clear from the requirement nmV^{∘} = nV^{∘}. q.e.d.

To prove Theorem B, we use the following theorem on the punctured local holomorphic de Rham cohomology in the hypersurface case.

Theorem 3.8 ([3]). Let (V, 0) = {(z_{0}, …, z_{n}) ∈ ℂ^{n+1} : f(z_{0}, …, z_{n}) = 0} be a hypersurface with the origin as an isolated singular point. Then:

(1) dim H_{h}^{k}(V, 0) = 0 for 1 ⩽ k ⩽ n − 2.

(2) dim H_{h}^{n}(V, 0) − dim H_{h}^{n−1}(V, 0) = μ − τ, where

μ = dim ℂ\{z_{0}, …, z_{n}\}/ (\frac{∂f}{∂z_{0}} , …, \frac{∂f}{∂z_{n}} )

is the Milnor number and

τ = dim ℂ\{z_{0}, …, z_{n}\}/ (f, \frac{∂f}{∂z_{0}} , …, \frac{∂f}{∂z_{n}} )

is the Tjurina number of the singularity (V, 0) respectively.

In particular, dim H_{h}^{n}(V, 0) = dim H_{h}^{n−1}(V, 0) if and only if (V, 0) is a quasi-homogeneous singularity, i.e., f is a weighted homogeneous polynomial after a holomorphic change of coordinates.

Proof of Theorem B. In view of Corollary 2.7 of [5] we know that the hypersurface singularity (V, x) is quasi-homogeneous. Then Theorem B follows from Theorem A and Theorem 3.8. q.e.d.

Let X ⊂ ℂ^{N} be a compact connected strongly pseudoconvex 3dimensional CR manifold with a transversal holomorphic S^{1}-action. In view of Theorem 2.1 of [5], there exists a holomorphic equivariant embedding X ↪ Ṽ as a real hypersurface in a 2-dimensional algebraic variety Ṽ ⊂ ℂÑ with a linear ℂ^{∗}-action; moreover, the linear ℂ^{∗}-action has exactly one fixed point x, which is the only possible singularity of Ṽ. We assume here that the fixed point x is a rational singularity. By Theorem A, H_{h}^{k}(X) = H_{h}^{k}(V, x) where V is the compact subset of Ṽ bounded by X.

We now recall the following important result of Campana and Flenner.

Proposition 3.9 ([2]). If (V, 0) is rational isolated singularity of dimension n ⩾ 2, then any closed holomorphic p-form η on V \{0} with 1 ⩽ p ⩽ 2 is exact, i.e., after shrinking V as a neighborhood of 0, there exists a (p − 1)-form ξ on V \ {0} with d(ξ) = η.

Proof of Theorem C. By Theorem A, H_{h}^{k}(X) ≅ H_{h}^{k}(V, x), for k = 1,

2. The above proposition of Campana and Flenner says that H_{h}^{k}(V, x) = 0, k = 1, 2. q.e.d.

Proof of Theorem D. Let x be the singularity of V. Then (V, x) is an isolated quasi-homogeneous hypersurface singularity. If H_{h}^{2}(X) = 0, then Theorem A of [6] implies that dim H_{h}^{2}(M) = 0 = s, where s = dim Γ(M \ A, Ω^{2})/(dΓ(M \ A, Ω^{1}) + Γ(M, Ω^{2})) is an invariant of (V, x). Vanishing of s implies H^{1}(A, ℂ) = 0 by Theorem B of [6].

Conversely if H^{1}(A, ℂ) = 0, then Theorem B of [6] implies s = 0 and hence H_{h}^{1}(M) = 0 = H_{h}^{2}(M). In view of Theorem A of [6], we have H_{h}^{1}(X) ≅ H_{h}^{1}(M) = 0 and dim H_{h}^{2}(X) = dim H_{h}^{2}(M) + s = 0. q.e.d.

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The Chinese University of Hong Kong Shatin, N.T., Hong Kong

East China Normal University Shanghai 200062, China, and

University of Illinois at Chicago Chicago, IL 60607-7045