Quaternionic Dolbeault complex and vanishing theorems on hyperkahler manifolds

Abstract

Let (M,I,J,K) be a hyperkahler manifold of real dimension 4n, 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 the Chern class c_1(L) lies in the closure $\hat{K}$ of the dual Kahler cone, then $H^i(L)=0$ for i>n. If c_1(L) lies in the opposite cone $-\hat{K}$, then $H^i(L)=0$ for i<n. Finally, if $c_1(L)$ is neither in $\hat{K}$ nor in $-\hat{K}$, then $H^i(L)=0$ for $i\ne n$.

30 pages, version 2 - misprints corrected, some arguments improved

References in corpus (1)

• Hyperholomorpic connections on coherent sheaves and stability