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## Positive forms on hyperkahler manifolds

Osaka Journal of Mathematics. 2010. Vol. 47. No. 2. P. 353-384.

Let (M; I; J;K; g) be a hyperkahler manifold, dimRM = 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 : 2p;0
I (M) !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 hyperkahler 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 hyperkahler version of the Sibony's lemma, showing
that a closed, positive (2p; 0)-form dened 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.

Research target:
Philosophy, Ethics, and Religious Studies

Language:
English