Stability of Gieseker stable sheaves on K3 surfaces in the sense of Bridgeland and some applications

TL;DR

This paper demonstrates that certain Gieseker stable sheaves on K3 surfaces are also stable under Bridgeland stability conditions, and explores implications for moduli spaces of sheaves, especially when the Picard number is one.

Contribution

It establishes conditions under which Gieseker stable sheaves are Bridgeland stable on K3 surfaces and applies this to describe moduli spaces of sheaves.

Findings

Gieseker stable sheaves can be Bridgeland stable in explicit stability condition subsets.

The moduli space of Gieseker stable torsion free sheaves coincides with that of μ-stable locally free sheaves when rank is not a perfect square.

Provides criteria linking Gieseker and Bridgeland stability for sheaves on K3 surfaces.

Abstract

We show that some Gieseker stable sheaves on a projective K3 surface $X$ are stable with respect to a stability condition of Bridgeland on the derived category of $X$ if the stability condition is in explicit subsets of the space of stability conditions depending on the sheaves. Furthermore we shall give two applications of the result. As a part of these applications, we show that the fine moduli space of Gieseker stable torsion free sheaves on a K3 surface with Picard number one is the moduli space of $\mu$-stable locally free sheaves if the rank of the sheaves is not a square number.