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

Proceedings of The Eleventh International Workshop on Diﬀ. Geom. 11(2007) 17-24

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 bundle E and 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. Let M be a compact, strongly pseudo convex CR (2n−1)-dimensional manifold and E be a holomorphic vector bundle over M. Assume 2n − 1 ≥ 5 in what follows. Then we have Theorem A([I-S]). Let Hp,q(M; E) be the space of E-valued harmonic (p, q)-forms over M. Then

Hp,q(M; E) =∼ Hn−p,n−q−1(M; E∗)(1.1) for any (p, q), 0 ≤ p ≤ n, 0 ≤ q ≤ n − 1. Here E∗ is the dual of E.

q p ˆ∗ Theorem B([I-S]). The q-th cohomology group H (M; E ⊗ Λ TM ) satisﬁes q p ˆ∗ ∼ n−q−1 ∗ n−p ˆ∗ H (M; E ⊗ Λ TM ) = H (M; E ⊗ Λ TM )(1.2) for any (p, q), 0 ≤ p ≤ n, 0 ≤ q ≤ n − 1. Here TˆM is the holomorphic tangent bundle of M and Hq(M; F ) denotes the ∂-cohomology group of the ∂-complex q {Cq(M; F ); ∂ = ∂ } associated with a holomorphic vector bundle F . Moreover, q ∼ n−q−1 ∗ (1.3) H (M; E) = H (M; E ⊗ KˆM ), 0 ≤ q ≤ n − 1, ˆ n ˆ∗ where KM = Λ TM denotes the canonical complex line bundle of M. 1.2 Over a compact complex n-dimensional complex manifold M the following duality

Hq(M n;Ωp(F )) =∼ Hn−q(M;Ωn−p(F ∗))(1.4)

17 18 Mitsuhiro Itoh holds for a holomorphic vector bundle F and 0 ≤ p, q ≤ n, where Ωp(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:

(1.5) Hp,q(M) =∼ Hn−p,n−q−1(M), for any (p, q), 0 ≤ p ≤ n, 0 ≤ q ≤ n − 1. Here Hp,q(M) is the space of harmonic (p, q)-forms on M.

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 call M or M 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 TM deﬁned by P = Ker θ, (ii) S is a complex subbundle of the tangent vector bundle T CM such that

S ∩ S = {0}, and

[Γ(S), Γ(S)] ⊂ Γ(S)(2.6)

(iii) Further

P C = S ⊕ S

2 and I is an endomorphism of P satisfying I = −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 metric h = g ⊕ θ ⊗ θ on M. Since a s.p.c. CR manifold M is contact, M admits a smooth vector field ξ, called basic field ( or Reeb field). The Serre Duality for Holomorphic Vector Bundles over Strongly Pseudo Convex CR Manifolds 19

We call a s.p.c. CR manifold normal, when it holds (2.8) [ξ, Γ(S)] ⊂ Γ(S).

2.2 Example 1. The following are typical examples of s.p.c. CR manifolds. (a) a strictly convex real hypersurface M in a complex manifold N, (b) a Sasakian manifold, (c) an S1-bundle over a complex Hodge manifold, (d) link of zero locus of a complex weighted homogenious polynomial

X a1 an+1 f(z1, ··· , zn+1) = ci1i2···in+1 z1 ··· zn+1 ,

namely, the link Kf is deﬁned 2n+1 n+1 (2.9) Kf = S ∩ {z ∈ C | f(z) = 0}. Notice that examples (b), (c) and (d) are normal s.p.c. manifolds

3 Holomorphic Vector Bundles

3.1 Let E −→ M be a complex vector bundle over a s.p.c. CR manifold M. Definition. A holomorphic structure of E is a ﬁrst order diﬀerential operator ∗ ∂ = ∂E : Γ(M; E) −→ Γ(M; E ⊗ S ) satisfying (3.10) (i) ∂(fu) = f∂(u) + u ⊗ d”f, (here d” denotes the CR operator, that is, the S-component of the ordinary exterior derivative d), namely, if we set ∂X (u) = ∂(u)(X), X ∈ S, then, equivalently 0 (3.11) (i ) ∂X (fu) = f∂X (u) + Xf u, and

(3.12) (ii) ∂X ∂Y u − ∂Y ∂X u − ∂[X,Y ]u = 0, u ∈ Γ(M; E) and X,Y are smooth sections of S. We call the bundle (E, ∂), or E holomorphic. The condition (ii) means the vanishing of (0, 2)-component of the curvature form RD, when E admits a connection D whose (0, 1)-part is the operator ∂. A smooth section u of E is called holomorphic when it satisﬁes ∂u = 0. C C 3.2 Example 2. The quotient complex vector bundle TˆM = T M/S of T M is holomorphic. It is called the holomorphic tangent bundle of M, when M is s.p.c. CR. The holomorphic structure ∂ is deﬁned

(3.13) ∂X u = π([X,Z]), 20 Mitsuhiro Itoh

C C where Z is a section of T M such that π(Z) = u and π : T M −→ TˆM is the projection. Notice. If M is normal, then the basic ﬁeld ξ, more precisely π(ξ), is a holomorphic section of TˆM 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 ΛkE are holomorphic. Furthermore, if U is a holomorphic subbundle of a holomorphic vector bundle E, 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 and M be a s.p.c. CR manifold whose CR structure comes from the S1-bundle over M. Then it is not hard to show that the pull-back of E over M is holomorphic. Example 4. If M is a s.p.c. CR manifold M which is a real hypersurface in a complex manifold M, then any holomorphic vector bundle E over M restricts to M holomorphic. Notice that if dim M ≥ 7, any holomorphic vector bundle E over a s.p.c. CR manifold fulﬁlls the local holomorphic frame property, i.e., around any point of M E admits a local holomorphic frame (s1, ··· , sr). It still open whether this property holds even when dim ≤ 5.

4 Cohomologies Hp,q(M; E)

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

q (∂ ϕ)(Y 1, ··· , Y q+1)(4.15) X = (−1)j+1∂ (ϕ(Y , ··· , Yˆ , ··· , Y )) Y j 1 j q+1 j X i+j ˆ ˆ + (−1) ϕ([Y i, Y j], Y 1, ··· , Y i, ··· , Y j, ··· , Y q+1), i,j

q where ϕ ∈ C (M; E),Yj ∈ Γ(M; S), j = 1, ··· , q + 1. It holds

q+1 q (4.16) ∂ ◦ ∂ = 0. The Serre Duality for Holomorphic Vector Bundles over Strongly Pseudo Convex CR Manifolds 21

Hence we have the cohomology group Hq(M; E), q = 0, ··· , n − 1 as

q q−1 (4.17) Hq(M; E) = Ker ∂ / Im ∂ .

4.2 To deﬁne the (p, q)-cohomology group Hp,q(M; E) we take the holomorphic p ˆ∗ vector bundle E ⊗ Λ (TM ) by tensoring E with the holomorphic exterior product p ˆ∗ p,q p ˆ∗ q ∗ bundle Λ (TM ), and by C (M; E) we denote Γ(M; E ⊗ Λ TM ⊗ Λ S ), the space p ˆ∗ q ∗ of all smooth sections of E ⊗ Λ TM ⊗ Λ S . q 0,q ∼ Notice that in our terminology C (M; E) = C (M; E). Further TˆM = Cξ ⊕ S ˆ∗ ∼ ∗ as a smooth complex vector bundle so that TM = Cθ ⊕ S and p ˆ∗ ∼ p−1 ∗ p p (4.18) Λ TM = Cθ ⊗ Λ S ⊕ Λ S .

Definition. The (p, q)-cohomology Hp,q(M; E) is deﬁned p,q q p ˆ∗ (4.19) H (M; E) = H (M; E ⊗ Λ TM ).

5 Harmonic E-valued (p, q)-forms

5.1 Let M be, same as before, a s.p.c. CR manifold. Then M admits the canonical volume form dv = θ∧(dθ)n−1 and also the Riemannian metric h = g ⊕θ⊗θ induced from the Levi form g and the contact form θ of M. p ˆ∗ In order to get the notion of harmonic forms taking values in E ⊗ Λ TM we ∗ q need to define the formal adjoint ∂ of the operator ∂ = ∂. For this, just like the complex manifold case as in [K-M], we define the Hodge star operator p ˆ∗ q ∗ ∗ n−p ˆ∗ n−q−1 ∗ (5.20) ] : E ⊗ Λ TM ⊗ Λ S −→ E ⊗ Λ TM ⊗ Λ S , defined by ψ −→ ](ψ) = ∗ψ, by exploiting the identification (18). Then the formal ∗ adjoint ∂ is defined ∗ (5.21) ∂ = (−1)n−k]∗ ◦ ∂ ◦ ]

p ˆ∗ q ∗ ∗ 2 on E ⊗ Λ TM ⊗ Λ S , k = p + q. It is not diﬃcult to show that ∂ is the L -adjoint of the holomorphic operator ∂ : ∗ (5.22) h∂ϕ, ψi = hϕ, ∂ ψi for ϕ ∈ Cp,q(M; E) and ψ ∈ Cp,q+1(M; E∗). The inner product is Z hϕ, ψi = (ϕ, ψ) dv. M

A section ϕ of Cp,q(M; E) is called harmonic when it satisfies ∗ ∗ (∂ ∂ + ∂∂ )ϕ = 0(5.23) 22 Mitsuhiro Itoh and denote by Hp,q(M; E) the space of all E-valued harmonic (p, q)-forms. 5.2 By Kohn’s theorem([K]), if dim M ≥ 5, then Hp,q(M; E) is finite dimensional and Cp,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 Theorem B from Theorem A. Proof of Theorem A. Let ψ be an element of Hp,q(M; E). Then we have by definition

∗ ∂Eψ = 0, and ∂Eψ = 0.

q p ˆ∗ Since ψ ∈ C (M; E ⊗ Λ TM ),

n−q−1 ∗ n−p ˆ∗ (5.24) ] ψ ∈ C (M; E ⊗ Λ TM ).

Note that ]ψ is an E∗-valued form. Then by exploiting the deﬁnition of the formal adjoint, we have

n−k ∗ ∂E∗ (]ψ) = (−1) ] (∂E) ψ = 0, where k = p + q and

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

Thus, we have

]ψ ∈ Hn−p,n−q−1(M; E∗)(5.25) and from that ]∗ = ]−1, ] induces the isomorphism

] : Hp,q(M; E) −→ Hn−p,n−q−1(M; E∗)(5.26) to obtain Theorem A.

6 Remarks

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

Hq(M; E) =∼ H0,q(M; E)(6.27) =∼ Hn,n−q−1(M; E∗) ∼ 0,n−q−1 n ˆ∗ ∗ = H (M;Λ TM ⊗ E ) ∼ n−q−1 n ˆ∗ ∗ = H (M;Λ TM ⊗ E ),

n−q−1 ∗ which is just H (M; E ⊗ KˆM ). The Serre Duality for Holomorphic Vector Bundles over Strongly Pseudo Convex CR Manifolds 23

6.2 Let M be a s.p.c. CR (2n − 1)-dimensional manifold which is normal and ξ be a basic field of M. Then ξ is a holomorphic section of the holomorphic tangent bundle TˆM which vanishes nowhere. Then this field ξ induces the inner product which satisfies for q p ˆ∗ p ˆ∗ q ∗ α ∈ C (M; E ⊗ Λ TM ) = Γ(M; E ⊗ Λ TM ⊗ Λ S )

(6.28) ∂ iξα = iξ ∂α so that this induces a linear map between the cohomology groups:

p,q p−1,q iξ : H (M; E) −→ H (M; E):[ψ] 7→ [iξψ](6.29) which fullﬁls

(6.30) iξ ◦ iξ = 0.

4 6.3 Example 5 Let M = Kf be a link of zero locus in C of the weighted 6 6 6 2 homogeneous polynomial f = z1 + z2 + z3 + z4 . Then dim M = 5 and the weight of f is (1, 1, 1, 3) and the degree d = 6. So the Hodge numbers are h2,0(M) = 1 and h1,1(M) = 19 by computing in terms of the Milnor algebra associated to the p,q p,q p,q f, where h (M) = dimC H (M). See [I] for the counting formula of h (M). By applying the Serre duality in the trivial bundle case, M admits a non-trivial 0 3 ˆ∗ ∼ holomorphic 3-form ψ, since H (M;Λ TM ) = C. This is because 2,0 0,2 3,0 3,0 h = h = h = dimC H (M). Here the second equality is from the Serre duality and then 3,0 ∼ 0 3 ˆ∗ H (M) = H (M;Λ TM ). 0 2 ˆ∗ Moreover iξψ ∈ H (M;Λ TM ) is a holomorphic 2-form on M which is also non-trivial so that this form yields a base of H2,0(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. 24 Mitsuhiro Itoh

[O-S-S] C. Okonek, M. Schneider and H. Spindler, Vector Bundles on Complex Projective Spaces, Birkhäuser, 1980. [S] J.-P. Serre, Une théorèmde dualité,Comm. Math. Helv., 29, 1-197, 1940. [T] N. Tanaka, A Differential Geometric Study on Strongly Pseudo-Convex Mani- fold, Kinokuniya, Tokyo, 1975.