Positive forms on hyperkähler manifolds

Abstract

Let $(M,I,J,K,g)$ be a hyperkähler manifold, $\dim_{\mathbb{R}} M =4n$. We study positive, $\partial$-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 $\mathcal{V}_{p,p}\colon \Lambda^{2p,0}_{I}(M)\to\Lambda^{n+p,n+p}_{I}(M)$, which maps $\partial$-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 $Z\subset (M,I)$, $\codim Z > 2p$ is locally integrable in a neighbourhood of $Z$. These results are used to prove polystability of derived direct images of certain coherent sheaves.