Cusps of the K\"ahler moduli space and stability conditions on K3 surfaces

TL;DR

This paper explores the relationship between cusps of the K"ahler moduli space of K3 surfaces and stability conditions, establishing connections with Fourier--Mukai partners and moduli spaces of stable objects.

Contribution

It investigates the link between stability conditions near cusps and Fourier--Mukai partners, comparing hearts of stability conditions and constructing moduli spaces of stable objects.

Findings

Comparison of hearts of stability conditions and coherent sheaves on Y

Construction of Y as a moduli space of stable objects

Analysis of auto-equivalences respecting the stability component

Abstract

In [Ma1] S. Ma established a bijection between Fourier--Mukai partners of a K3 surface and cusps of the K\"ahler moduli space. The K\"ahler moduli space can be described as a quotient of Bridgeland's stability manifold. We study the relation between stability conditions $\sigma$ near to a cusp and the associated Fourier--Mukai partner Y in the following ways. (1) We compare the heart of $\sigma$ to the heart of coherent sheaves on Y. (2) We construct Y as moduli space of $\sigma$-stable objects. An appendix is devoted to the group of auto-equivalences of the derived category which respect the component $Stab^\dagger(X)$ of the stability manifold.