Lagrangian Fibrations of Holomorphic-Symplectic Varieties of K3[n]-Type

By E. Markman. 2013

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Let X be a compact Kahler holomorphic-symplectic manifold, which is deformation equivalent to the Hilbert scheme of length n subschemes of a K3 surface. Let \(\mathcal{L}\) be a nef line-bundle on X, such that the top power \(c_{1}(\mathcal{L})^{2n}\) vanishes and \(c_{1}(\mathcal{L})\) is primitive. Assume that the two dimensional subspace H 2, 0(X) ⊕ H 0, 2(X) of \(H^{2}(X, \mathbb{C})\) intersects \(H^{2}(X, \mathbb{Z})\) trivially. We prove that the linear system of \(\mathcal{L}\) is base point free and it induces a Lagrangian fibration on X. In particular, the line-bundle \(\mathcal{L}\) is effective. A determination of the semi-group of effective divisor classes on X follows, when X is projective. For a generic such pair \((X,\mathcal{L})\), not necessarily projective, we show that X is bimeromorphic to a Tate-Shafarevich twist of a moduli space of stable torsion sheaves, each with pure one dimensional support, on a projective K3 surface.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9f30825eecc3b877c8f42f49e08cb8cf132049be