Quaternionic Dolbeault complex and vanishing theorems on hyperkahler manifolds

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 $\widehat{K}$ of the dual Kahler cone, then $H^i(L)=0$ for i>n. If c_1(L) lies in the opposite cone $-\widehat{K}$, then $H^i(L)=0$ for i<n. Finally, if $c_1(L)$ is neither in $\widehat{K}$ nor in $-\widehat{K}$, then $H^i(L)=0$ for $i\mathrm{\ne }n$.