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 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.

ABSTRACT. We construct new t-structures on the derived category of coherent sheaves on smooth projective threefolds. We conjecture that they give Bridgeland stability conditions near the large volume limit. We show that this conjecture is equivalent to a BogomolovGieseker type inequality for the third Chern character of certain stable complexes. We also conjecture a stronger inequality, and prove it in the case of projective space, and for various examples.Finally, we prove a version of the classical Bogomolov-Gieseker inequality, not involving the third Chern character, for stable complexes.

We construct a family of nef divisor classes on every moduli space of stable complexes in the sense of Bridgeland. This divisor class varies naturally with the Bridgeland stability condition. For a generic stability condition on a K3 surface, we prove that this class is ample, thereby generalizing a result of Minamide, Yanagida, and Yoshioka. Our result also gives a systematic explanation of the relation between wall-crossing for Bridgeland-stability and the minimal model program for the moduli space.We give three applications of our method for classical moduli spaces of sheaves on a K3 surface:1. We obtain a region in the ample cone in the moduli space of Gieseker-stable sheaves only depending on the lattice of the K3.2. We determine the nef cone of the Hilbert scheme of n points on a K3 surface of Picard rank one when n is large compared to the genus.3. We verify the "Hassett-Tschinkel/Huybrechts/Sawon" conjecture on the existence of a birational Lagrangian fibration for the Hilbert scheme in a new family of cases.