Stability conditions via spherical objects

TL;DR

This paper demonstrates that the stability condition on the derived category of a K3 surface can be fully determined by the stability of its spherical objects, revealing their fundamental role in the structure of the category.

Contribution

It establishes that spherical objects uniquely determine stability conditions in the derived category of a K3 surface, highlighting their importance in understanding the category's structure.

Findings

Spherical objects are key to understanding stability conditions.

Stability conditions are determined by spherical objects.

Spherical objects influence the structure of the derived category.

Abstract

An object in the bounded derived category D^b(Coh(X)) of coherent sheaves on a complex projective K3 surface X is spherical if it is rigid and simple. Although spherical objects form only a discrete set in the moduli stack of complexes, they determine much of the structure of X and D^b(Coh(X)). Here we show that a stability condition on D^b(Coh(X)) is determined by the stability of spherical objects.