Osaka Journal of Mathematics

Positive forms on hyperkähler manifolds

Misha Verbitsky

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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.

Article information

Osaka J. Math., Volume 47, Number 2 (2010), 353-384.

First available in Project Euclid: 23 June 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32F17: Other notions of convexity 53C26: Hyper-Kähler and quaternionic Kähler geometry, "special" geometry 32U05: Plurisubharmonic functions and generalizations [See also 31C10]


Verbitsky, Misha. Positive forms on hyperkähler manifolds. Osaka J. Math. 47 (2010), no. 2, 353--384.

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