  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 $i\neq n$.
