Bridgeland-Stable Moduli Spaces for K-Trivial Surfaces

TL;DR

This paper constructs and analyzes Bridgeland stability conditions on K-trivial surfaces, describing wall-crossing phenomena and constructing moduli spaces of stable objects, with applications to K3 and Abelian surfaces.

Contribution

It introduces a natural family of Bridgeland stability conditions on K-trivial surfaces and constructs associated moduli spaces using Mukai flops and elementary modifications.

Findings

Wall-crossing behavior for objects similar to $\\cO_C(H)$

Construction of fine moduli spaces via Mukai flops

Generalization of Thaddeus' stable pairs

Abstract

We give a natural family of Bridgeland stability conditions on the derived category of a smooth projective complex surface S and describe ``wall-crossing behavior'' for objects with the same invariants as $\cO_C(H)$ when H generates Pic(S) and $C \in |H|$. If, in addition, S is a K3 or Abelian surface, we use this description to construct a sequence of fine moduli spaces of Bridgeland-stable objects via Mukai flops and generalized elementary modifications of the universal coherent sheaf. We also discover a natural generalization of Thaddeus' stable pairs for curves embedded in the moduli spaces.