MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations

Abstract

We use wall-crossing with respect to Bridgeland stability conditions to systematically study the birational geometry of a moduli space M of stable sheaves on a K3 surface X: 1. We describe the nef cone, the movable cone, and the effective cone of M in terms of the Mukai lattice of X. 2. We establish a long-standing conjecture that predicts the existence of a birational Lagrangian fibration on M whenever M admits an integral divisor class D of square zero (with respect to the Beauville-Bogomolov form). These results are proved using a natural map from the space of Bridgeland stability conditions Stab(X) to the cone Mov(X) of movable divisors on M; this map relates wall-crossing in Stab(X) to birational transformations of M. In particular, every minimal model of M appears as a moduli space of Bridgeland-stable objects on X.

62 pages, 5 figures. Comments welcome. v2: fixed sign issue in statement of Theorem 12.2. v3: strengthened Corollary 1.3. Many improvements and corrections to the exposition, based on the referee's comments. Published version. v4: Section 13: fixed two typos in published version (see footnotes 5 and 6); clarified (33)

References in corpus (2)

• Base manifolds for fibrations of projective irreducible symplectic manifolds
• Webs of Lagrangian Tori in Projective Symplectic Manifolds

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• The Effective Cone of Moduli Spaces of Sheaves on a Smooth Quadric Surface
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• Symplectic involutions of $K{3}^{[n]}$ type and Kummer $n$ type manifolds
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