Positive forms on hyperkähler manifolds

Let (M, I, J,K, g) be a hyperkähler manifold, dimR M = 4n. We study positive, ∂-closed (2p, 0)-forms on (M, I). These forms are quaternionic analogues of the positive (p, p)-forms, well-known in complex geometry. We construct a monomorphism Vp,p : Λ I (M)−→ Λ n+p,n+p I (M), which maps ∂-closed (2p, 0)-forms to closed (n+p, n+p)-forms, and positive (2p, 0)forms to positive (n + p, n + p)-forms. This construction is used to prove a hyperkähler version of the classical Skoda-El Mir theorem, which says that a trivial extension of a closed, positive current over a pluripolar set is again closed. We also prove the hyperkähler version of the Sibony’s lemma, showing that a closed, positive (2p, 0)-form defined outside of a compact complex subvariety Z ⊂ (M, I), codimZ > 2p is locally integrable in a neighbourhood of Z. These results are used to prove polystability of derived direct images of certain coherent sheaves.