The Serre Duality for Holomorphic Vector Bundles over Strongly Pseudo Convex CR Manifolds

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The Serre Duality for Holomorphic Vector Bundles over Strongly Pseudo Convex CR Manifolds

Mitsuhiro Itoh

Institute of Mathematics, University of Tsukuba, Japan e-mail: [email protected]

(2000 Mathematics Subject Classification : 53C40, 53C15. )

1 Main Theorem.

1.1 The Serre duality for a holomorphic vector bundleEand its dual vector bundle E^∗ holds over a compact, strongly pseudo convex CR manifold. This duality is a vector bundle generalization of the Serre duality for scalar valued harmonic forms over a strongly pseudo convex CR manifold given by N. Tanaka.

LetM be a compact, strongly pseudo convex CR (2n−1)-dimensional manifold andE be a holomorphic vector bundle overM. Assume 2n−1≥5 in what follows.

Then we have

Theorem A([I-S]). LetH^p,q(M;E) be the space ofE-valued harmonic (p, q)-forms overM. Then

H^p,q(M;E)∼=H^{n−p,n−q−1}(M;E^∗) (1.1)

for any (p, q), 0≤p≤n, 0≤q≤n−1. HereE^∗ is the dual ofE.

TheoremB([I-S]). Theq-th cohomology groupH^q(M;E⊗Λ^pTˆ_M^∗) satisfies H^q(M;E⊗Λ^pTˆ_M^∗)∼=H^n−q−1(M;E^∗⊗Λ^n−pTˆ_M^∗)

(1.2)

for any (p, q), 0 ≤ p ≤ n, 0 ≤ q ≤ n−1. Here ˆTM is the holomorphic tangent bundle of M and H^q(M;F) denotes the ∂-cohomology group of the ∂-complex {C^q(M;F);∂=∂^q}associated with a holomorphic vector bundleF.

Moreover,

H^q(M;E)∼=H^n−q−1(M;E^∗⊗KˆM), 0≤q≤n−1, (1.3)

where ˆK_M = ΛⁿTˆ_M^∗ denotes the canonical complex line bundle ofM.

1.2 Over a compact complex n-dimensional complex manifold M the following duality

H^q(Mⁿ; Ω^p(F))∼=H^n−q(M; Ω^n−p(F^∗)) (1.4)

17

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holds for a holomorphic vector bundle F and 0≤ p, q ≤n, where Ω^p(F) denotes the sheaf of holomorphic F-valued p-forms. This is called Serre duality due to J.P. Serre ([S]), which is a fundamental machinery in study of the ∂-cohomology of holomorphic vector bundles over a complex manifold. See for this [K] and also [O-S-S], for examples.

1.3 In his famous lecture note [T] N.Tanaka established the following duality over a strongly pseudo convex CR manifold in scalar valued harmonic (p, q)-forms:

H^p,q(M)∼=H^{n−p,n−q−1}(M), (1.5)

for any (p, q), 0≤p≤n, 0 ≤q≤n−1. Here H^p,q(M) is the space of harmonic (p, q)-forms onM.

2 Strongly Pseudo Convex CR Manifolds

2.1 Let M be a (2n−1)-dimensional manifold. If M carries the following CR structure (S, θ, P, I, g), we callM orM with (S, θ, P, I, g) a strongly pseudo convex CR manifold (by abbrevation, s.p.c. CR manifold);

(i) θ is a contact form of M so that P is a subbundle of T M defined by P = Kerθ,

(ii) S is a complex subbundle of the tangent vector bundleT^CM such that S∩S={0},

and

[Γ(S),Γ(S)]⊂Γ(S) (2.6)

(iii) Further

P^C=S⊕S

andI is an endomorphism ofP satisfyingI²=−idP and S={X∈P^C|IX=−√

−1X}

(iv) The symmetric bilinear form, called Levi form,g:P×P −→ R g(X, Y) =−dθ(IX, Y)

(2.7)

is positive definite. So the Levi form g together with the contact form θ yields a smooth Riemannian metrich=g⊕θ⊗θonM.

Since a s.p.c. CR manifold M is contact, M admits a smooth vector field ξ, called basic field ( or Reeb field).

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We call a s.p.c. CR manifold normal, when it holds [ξ,Γ(S)]⊂Γ(S).

(2.8)

2.2 Example 1. The following are typical examples of s.p.c. CR manifolds.

(a) a strictly convex real hypersurfaceM in a complex manifoldN, (b) a Sasakian manifold,

(c) anS¹-bundle over a complex Hodge manifold,

(d) link of zero locus of a complex weighted homogenious polynomial f(z1,· · ·, zn+1) =X

ci₁i₂···i_n+1z₁^a¹· · ·z_n+1^aⁿ⁺¹, namely, the linkK_f is defined

Kf =S²ⁿ⁺¹∩ {z∈Cⁿ⁺¹|f(z) = 0}.

(2.9)

Notice that examples (b), (c) and (d) are normal s.p.c. manifolds

3 Holomorphic Vector Bundles

3.1 LetE −→M be a complex vector bundle over a s.p.c. CR manifoldM. Definition. A holomorphic structure of E is a first order differential operator

∂=∂E : Γ(M;E)−→Γ(M;E⊗S^∗) satisfying (i) ∂(f u) =f ∂(u) +u⊗d”f, (3.10)

(hered” denotes the CR operator, that is, theS-component of the ordinary exterior derivatived), namely, if we set∂_X(u) =∂(u)(X),X∈S, then, equivalently

(i⁰) ∂_X(f u) =f ∂_X(u) +Xf u, (3.11)

and

(ii) ∂_X∂_Yu−∂_Y∂_Xu−∂_[X,Y_]u= 0, (3.12)

u∈Γ(M;E) and X, Y are smooth sections ofS. We call the bundle (E, ∂), orE holomorphic.

The condition (ii) means the vanishing of (0,2)-component of the curvature formR_D, whenE admits a connectionD whose (0,1)-part is the operator∂.

A smooth sectionuofE is calledholomorphicwhen it satisfies∂u= 0.

3.2 Example 2. The quotient complex vector bundle ˆT_M =T^CM/SofT^CM is holomorphic. It is called the holomorphic tangent bundle of M, whenM is s.p.c.

CR. The holomorphic structure∂ is defined

∂_Xu=π([X, Z]), (3.13)

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where Z is a section of T^CM such that π(Z) = u and π : T^CM −→ Tˆ_M is the projection.

Notice. IfM is normal, then the basic fieldξ, more preciselyπ(ξ), is a holomorphic section of ˆT_M

3.3 Similarly to holomorphic vector bundles over a complex manifold, we have the following basic facts.

The tensor product E ⊗F of holomorphic vector bundles E and F over a s.p.c. CR manifold is also holomorphic. Moreover the dual bundle E^∗ and the exterior product bundle Λ^kE are holomorphic. Furthermore, ifU is a holomorphic subbundle of a holomorphic vector bundleE, then it induces the quotient bundle E/U which is holomorphic;

0 −→ U −→ E −→ E/U −→ 0 (3.14)

Example 3. Let E −→M be a holomorphic vector bundle over a Hodge manifold M andM be a s.p.c. CR manifold whose CR structure comes from theS¹-bundle overM. Then it is not hard to show that the pull-back ofEoverM is holomorphic.

Example 4. If M is a s.p.c. CR manifold M which is a real hypersurface in a complex manifoldM, then any holomorphic vector bundleE overM restricts toM holomorphic.

Notice that if dimM ≥7, any holomorphic vector bundleE over a s.p.c. CR manifold fulfills the local holomorphic frame property, i.e., around any point ofM Eadmits a local holomorphic frame (s₁,· · ·, s_r). It still open whether this property holds even when dim≤5.

4 Cohomologies H^p,q(M;E)

4.1 Let E be a holomorphic vector bundle over a s.p.c. CR manifold M. We define a complex{C^q(M;E);∂=∂^q}, called the∂-complex as follows: C^q(M;E) = Γ(M;E⊗Λ^qS^∗),q= 0,1,· · ·n−1 and the operator∂^q : C^q(M;E)−→ C^q+1(M;E) by

(∂^q ϕ)(Y₁,· · ·, Y_q+1) (4.15)

= X

j

(−1)^j+1∂_Y

j(ϕ(Y1,· · · ,Yˆj,· · · , Yq+1))

+ X

i,j

(−1)^i+jϕ([Yi, Yj], Y1,· · ·,Yˆi,· · ·,Yˆj,· · ·, Yq+1),

whereϕ∈ C^q(M;E), Y_j ∈Γ(M;S), j= 1,· · · , q+ 1.

It holds

∂^q+1◦∂^q= 0.

(4.16)

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Hence we have the cohomology groupH^q(M;E),q= 0,· · ·, n−1 as H^q(M;E) = Ker∂^q/Im∂^q−1.

(4.17)

4.2 To define the (p, q)-cohomology group H^p,q(M;E) we take the holomorphic vector bundle E⊗Λ^p( ˆT_M^∗) by tensoring E with the holomorphic exterior product bundle Λ^p( ˆT_M^∗), and byC^p,q(M;E) we denote Γ(M;E⊗Λ^pTˆ_M^∗ ⊗Λ^qS^∗), the space of all smooth sections ofE⊗Λ^pTˆ_M^∗ ⊗Λ^qS^∗.

Notice that in our terminology C^q(M;E) =C^0,q(M;E). Further ˆTM ∼=Cξ⊕S as a smooth complex vector bundle so that ˆT_M^∗ ∼=Cθ⊕S^∗ and

Λ^pTˆ_M^∗ ∼=Cθ⊗Λ^p−1S^∗⊕Λ^pS^p. (4.18)

Definition. The (p, q)-cohomologyH^p,q(M;E) is defined H^p,q(M;E) =H^q(M;E⊗Λ^pTˆ_M^∗).

(4.19)

5 HarmonicE-valued (p, q)-forms

5.1 LetM be, same as before, a s.p.c. CR manifold. ThenM admits the canonical volume formdv=θ∧(dθ)ⁿ⁻¹and also the Riemannian metrich=g⊕θ⊗θinduced from the Levi formg and the contact formθofM.

In order to get the notion of harmonic forms taking values in E⊗Λ^pTˆ_M^∗ we need to define the formal adjoint∂^∗ of the operator∂^q =∂. For this, just like the complex manifold case as in [K-M], we define the Hodge star operator

] : E⊗Λ^pTˆ_M^∗ ⊗Λ^qS^∗ −→E^∗⊗Λ^n−pTˆ_M^∗ ⊗Λ^n−q−1S^∗, (5.20)

defined byψ−→](ψ) =∗ψ, by exploiting the identification (18). Then the formal adjoint∂^∗ is defined

∂^∗= (−1)^n−k]^∗◦∂◦] (5.21)

onE⊗Λ^pTˆ_M^∗ ⊗Λ^qS^∗,k=p+q. It is not difficult to show that∂^∗ is theL²-adjoint of the holomorphic operator∂:

h∂ϕ, ψi=hϕ, ∂^∗ψi (5.22)

forϕ∈ C^p,q(M;E) andψ∈ C^p,q+1(M;E^∗). The inner product is hϕ, ψi=

Z

M

(ϕ, ψ)dv.

A sectionϕofC^p,q(M;E) is calledharmonicwhen it satisfies (∂^∗∂+∂∂^∗)ϕ= 0

(5.23)

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and denote byH^p,q(M;E) the space of allE-valued harmonic (p, q)-forms.

5.2 By Kohn’s theorem([K]), if dimM ≥5, thenH^p,q(M;E) is finite dimensional andC^p,q(M;E) is endowed with the Hodge-decomposition property in terms of the Laplace operator ∂^∗∂+∂∂^∗, (for the details, see [T], or the original paper [K]), so that we get the first part of TheoremB from TheoremA.

Proof of Theorem A. Let ψ be an element of H^p,q(M;E). Then we have by definition

∂Eψ= 0, and ∂^∗_Eψ= 0.

Sinceψ∈ C^q(M;E⊗Λ^pTˆ_M^∗),

] ψ∈ C^n−q−1(M;E^∗⊗Λ^n−pTˆ_M^∗).

(5.24)

Note that ]ψis anE^∗-valued form.

Then by exploiting the definition of the formal adjoint, we have

∂E^∗(]ψ) = (−1)^n−k ](∂E)^∗ψ= 0, wherek=p+qand

∂^∗_E∗(]ψ) = (−1)^n−k ] ∂E]^∗(]ψ) = (−1)^n−k ] ∂Eψ= 0.

Thus, we have

]ψ∈H^{n−p,n−q−1}(M;E^∗) (5.25)

and from that]^∗=]⁻¹, ]induces the isomorphism ]:H^p,q(M;E)−→H^{n−p,n−q−1}(M;E^∗) (5.26)

to obtain TheoremA.

6 Remarks

6.1 To show the last half part of TheoremB, we observe the following.

H^q(M;E) ∼= H^0,q(M;E) (6.27)

∼= H^n,n−q−1(M;E^∗)

∼= H^0,n−q−1(M; ΛⁿTˆ_M^∗ ⊗E^∗)

∼= H^n−q−1(M; ΛⁿTˆ_M^∗ ⊗E^∗), which is justH^n−q−1(M;E^∗⊗KˆM).

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6.2 LetM be a s.p.c. CR (2n−1)-dimensional manifold which is normal and ξ be a basic field ofM. Thenξ is a holomorphic section of the holomorphic tangent bundle ˆTM which vanishes nowhere.

Then this fieldξinduces the inner product which satisfies for α∈ C^q(M;E⊗Λ^pTˆ_M^∗) = Γ(M;E⊗Λ^pTˆ_M^∗ ⊗Λ^qS^∗)

∂ iξα=iξ ∂α (6.28)

so that this induces a linear map between the cohomology groups:

i_ξ : H^p,q(M;E)−→H^p−1,q(M;E) : [ψ] 7→ [i_ξψ]

(6.29) which fullfils

iξ◦iξ = 0.

(6.30)

6.3 Example 5 Let M = Kf be a link of zero locus in C⁴ of the weighted homogeneous polynomialf =z₁⁶+z⁶₂+z⁶₃+z²₄. Then dimM = 5 and the weight of f is (1,1,1,3) and the degree d= 6. So the Hodge numbers are h^2,0(M) = 1 and h^1,1(M) = 19 by computing in terms of the Milnor algebra associated to the f, whereh^p,q(M) = dim_CH^p,q(M). See [I] for the counting formula of h^p,q(M).

By applying the Serre duality in the trivial bundle case,M admits a non-trivial holomorphic 3-formψ, sinceH⁰(M; Λ³Tˆ_M^∗)∼=C. This is because

h^2,0=h^0,2=h^3,0= dim_CH^3,0(M).

Here the second equality is from the Serre duality and then H^3,0(M)∼=H⁰(M; Λ³Tˆ_M^∗).

Moreover iξψ ∈ H⁰(M; Λ²Tˆ_M^∗) is a holomorphic 2-form on M which is also non-trivial so that this form yields a base ofH^2,0(M)∼=C.

References

[I] M. Itoh, Sasakian manifolds, Hodge decomposition and Milnor algebras, Kyushu J. Math., 58, 121-140, 2004.

[I-S] M. Itoh and T. Saotome, The Serre Duality for Holomorphic Vector Bundles over a Strongly Pseudo-Convex Manifold, to appear in Tsukuba J. Math., 2007.

[K] S. Kobayashi, Differential Geometry of Complex Vector Bundles, Iwanami, 1987.

[K] J.J. Kohn, Boundaries of complex manifolds, Proc. Conference on Complex Manifolds (Minneapolis), Springer-Verlag, New York, 1965.

[K-M] K. Kodaira and J. Morrow, Complex Manifolds, Holt, Rinehart and Win- ston, New York, 1970.

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[O-S-S] C. Okonek, M. Schneider and H. Spindler, Vector Bundles on Complex Projective Spaces, Birkh¨auser, 1980.

[S] J.-P. Serre, Une théorèm de dualité, Comm. Math. Helv., 29, 1-197, 1940.

[T] N. Tanaka, A Differential Geometric Study on Strongly Pseudo-Convex Mani- fold, Kinokuniya, Tokyo, 1975.