Quaternionic Dolbeault complex and vanishing theorems on hyperkähler manifolds

Abstract

AbstractLet (M,I,J,K) be a compact hyperkähler manifold, $\dim _{\mathbb {H}}M=n$, 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 $\hat K$ of the dual Kähler cone, then Hi(L)=0 for i>n. If c1(L) lies in the opposite cone $-\hat K$, then Hi(L)=0 for i<n. Finally, if c1(L) is neither in $\hat K$ nor in $-\hat K$, then Hi(L)=0 for $i\neq n$.