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

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.