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

TL;DR

This paper employs wall-crossing techniques with Bridgeland stability conditions to analyze the birational geometry of moduli spaces of sheaves on K3 surfaces, establishing descriptions of cones and confirming a conjecture on Lagrangian fibrations.

Contribution

It provides a systematic description of nef, movable, and effective cones of moduli spaces on K3s and proves a conjecture linking divisor classes to Lagrangian fibrations.

Findings

Describes nef, movable, and effective cones via Mukai lattice.

Establishes existence of Lagrangian fibrations for certain divisors.

Connects wall-crossing in stability conditions to birational geometry.

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.