Bridgeland-stable moduli spaces for $K$-trivial surfaces

Abstract

We give a one-parameter 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 ${\mathcal{O}}_C(H)$ 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.