Positive forms on hyperkahler manifolds

Abstract

Let $(M,I,J,K)$ be a hyperkaehler manifold, $\dim_\R M =4n$. We study positive, Dolbeault-closed $(2p,0)$-forms on $(M,I)$. These forms are quaternionic analogues of the positive $(p,p)$-forms. We construct an injective homomorphism mapping Dolbeault-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 hyperkaehler 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 hyperkaehler version of the Sibony's lemma, showing that a closed, positive $(2p,0)$-form defined outside of a compact complex subvariety $Z\subset (M,I)$, $\codim Z > 2p$ is locally integrable in a neighbourhood of $Z$. These results are used to prove polystability of derived direct images of certain coherent sheaves.