Formality conjecture for K3 surfaces

TL;DR

This paper proves the formality conjecture for K3 surfaces, showing that certain DG algebras associated with sheaves are formal, using the uniqueness of DG enhancements and extending results to Bridgeland stability.

Contribution

It provides a proof of the formality conjecture for K3 surfaces and extends it to derived objects with generic Bridgeland stability conditions.

Findings

DG algebra RHom(F,F) is formal for polystable sheaves on K3 surfaces

Formality extends to derived objects with generic Bridgeland stability

Utilizes uniqueness of DG enhancement of derived categories

Abstract

We give a proof of the formality conjecture of Kaledin and Lehn: on a complex projective K3 surface, the DG algebra RHom(F,F) is formal for any sheaf F polystable with respect to an ample line bundle. Our main tool is the uniqueness of DG enhancement of the bounded derived category of coherent sheaves. We also extend the formality result to derived objects that are polystable with respect to a generic Bridgeland stability condition.