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Статья

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 de ned 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.