Rigid currents on compact hyperkahler manifolds

Nessim Sibony; Andrey Soldatenkov; Misha Verbitsky

doi:10.24033/asens.2626

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arXiv · 2303.11362

Rigid currents on compact hyperkahler manifolds

Nessim Sibony·Andrey Soldatenkov·Misha Verbitsky

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Abstract

A rigid cohomology class on a complex manifold is a class that is represented by a unique closed positive current. The positive current representing a rigid class is also called rigid. For a compact Kahler manifold $X$ all eigenvectors of hyperbolic automorphisms acting on $H^{1,1}(X)$ that have non-unit eigenvalues are rigid classes. Such classes are always parabolic, namely, they belong to the boundary of the Kahler cone and have vanishing volume. We study parabolic $(1,1)$-classes on compact hyperkahler manifolds with $b_2 \geq 7$. We show that a parabolic class is rigid if it is not orthogonal to a rational vector with respect to the BBF form. This implies that a general parabolic class on a hyperkahler manifold is rigid.

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BibTeX RIS

Nessim Sibony, Andrey Soldatenkov, Misha Verbitsky. 2023-03-20. Rigid currents on compact hyperkahler manifolds. https://doi.org/10.24033/asens.2626

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