Rational curves on hyperkahler manifolds

Ekaterina Amerik; Misha Verbitsky

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arXiv · 1401.0479

Rational curves on hyperkahler manifolds

Ekaterina Amerik·Misha Verbitsky

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Abstract

Let $M$ be an irreducible holomorphically symplectic manifold. We show that all faces of the Kahler cone of $M$ are hyperplanes $H_i$ orthogonal to certain homology classes, called monodromy birationally minimal (MBM) classes. Moreover, the Kahler cone is a connected component of a complement of the positive cone to the union of all $H_i$. We provide several characterizations of the MBM-classes. We show the invariance of MBM property by deformations, as long as the class in question stays of type (1,1). For hyperkahler manifolds with Picard group generated by a negative class $z$, we prove that $\pm z$ is Q-effective if and only if it is an MBM class. We also prove some results towards the Morrison-Kawamata cone conjecture for hyperkahler manifolds.

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BibTeX RIS

Ekaterina Amerik, Misha Verbitsky. 2014-01-02. Rational curves on hyperkahler manifolds. https://arxiv.org/abs/1401.0479

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