Lagrangian fibrations of holomorphic-symplectic varieties of K3^[n]-type

TL;DR

This paper proves that under certain conditions, a nef line bundle on a holomorphic-symplectic manifold of K3^[n]-type induces a Lagrangian fibration, and describes the structure of such manifolds and their divisor classes.

Contribution

It establishes conditions under which a nef line bundle induces a Lagrangian fibration on K3^[n]-type manifolds and characterizes their structure via Tate-Shafarevich twists.

Findings

Line bundle L is base point free and induces a Lagrangian fibration.

Effective divisor classes are characterized for projective cases.

X is bimeromorphic to a Tate-Shafarevich twist of a moduli space of stable sheaves.

Abstract

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 L be a nef line-bundle on X, such that the 2n-th power of c_1(L) vanishes and c_1(L) is primitive. Assume that the two dimensional subspace H^{2,0}(X) + H^{0,2}(X), of the second cohomology of X with complex coefficients, intersects trivially the integral cohomology. We prove that the linear system of L is base point free and it induces a Lagrangian fibration on X. In particular, the line-bundle 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,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.