Vanishing theorems for locally conformal hyperkähler manifolds
نویسنده
چکیده
Let M be a compact locally conformal hyperkähler manifold. We prove a version of Kodaira-Nakano vanishing theorem for M . This is used to show that M admits no holomorphic differential forms, and the cohomology of the structure sheaf H(OM ) vanishes for i > 1. We also prove that the first Betti number of M is 1. This leads to a structure theorem for locally conformally hyperkähler manifolds, describing them in terms of 3-Sasakian geometry. Similar results are proven for compact Einstein-Weyl locally conformal Kähler manifolds.
منابع مشابه
Conformal mappings preserving the Einstein tensor of Weyl manifolds
In this paper, we obtain a necessary and sufficient condition for a conformal mapping between two Weyl manifolds to preserve Einstein tensor. Then we prove that some basic curvature tensors of $W_n$ are preserved by such a conformal mapping if and only if the covector field of the mapping is locally a gradient. Also, we obtained the relation between the scalar curvatures of the Weyl manifolds r...
متن کاملClassification of Bochner Flat Kähler Manifolds by Heisenberg, Spherical CR Geometry
A Bochner flat Kähler manifold is a Kähler manifold with vanishing Bochner curvature tensor. We shall give a uniformization of Bochner flat Kähler manifolds. One of the aims of this paper is to give a correction to the proof of our previous paper [9] concerning uniformization of Bochner flat Kähler manifolds. A Bochner flat locally conformal Kähler manifold is a locally conformal Kähler manifol...
متن کاملVanishing theorems on Hermitian manifolds
We prove the vanishing of the Dolbeault cohomology groups on Hermitian manifolds with ddc-harmonic Kähler form and positive (1, 1)-part of the Ricci form of the Bismut connection. This implies the vanishing of the Dolbeault cohomology groups on complex surfaces which admit a conformal class of Hermitian metrics, such that the Ricci tensor of the canonical Weyl structure is positive. As a coroll...
متن کاملQuaternionic Dolbeault complex and vanishing theorems on hyperkähler manifolds
Let (M, I, J,K) be a hyperkähler manifold, dimH M = n, and L a non-trivial holomorphic line bundle on (M, I). Using the quaternionic Dolbeault complex, we prove the following vanishing theorem for holomorphic cohomology of L. If c1(L) lies in the closure K̂ of the dual Kähler cone, then H(L) = 0 for i > n. If c1(L) lies in the opposite cone −K̂, then H(L) = 0 for i < n. Finally, if c1(L) is neith...
متن کاملHyperkähler manifolds with torsion obtained from hyperholomorphic bundles
We construct examples of compact hyperkähler manifolds with torsion (HKT manifolds) which are not homogeneous and not locally conformal hyperkähler. Consider a total space T of a tangent bundle over a hyperkähler manifold M . The manifold T is hypercomplex, but it is never hyperkähler, unless M is flat. We show that T admits an HKT-structure. We also prove that a quotient of T by a Z-action v −...
متن کامل