Quaternionic Dolbeault complex and vanishing theorems on hyperkahler manifolds

Let (M, I, J, K) be a compact hyperkahler manifold, dim(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 c(1) (L) lies in the closure (K) over cap of the dual Kahler cone, then H-i(L) = 0 for i > n. If c(1)(L) lies in the opposite cone -(K) over cap, then H-i(L) = 0 for i < n. Finally, if c(1)(L) is neither in <(K)over cap> nor in -(K) over cap, then H-i(L) = 0 for i 7 n.