MMP FOR MODULI OF SHEAVES ON K3S VIA WALL-CROSSING: NEF AND MOVABLE CONES, LAGRANGIAN FIBRATIONS

Abstract

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: (a) We describe the nef cone, the movable cone, and the effective cone of M in terms of the Mukai lattice of X. (b) 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 coneMov(X) of movable divisors onM; 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. CONTENTS