Deformations of polystable sheaves on surfaces: quadraticity implies formality

TL;DR

This paper explores the connection between the quadraticity of the Kuranishi family of a sheaf and the formality of its derived endomorphism DG-Lie algebra, establishing an equivalence on smooth projective surfaces.

Contribution

It proves that for polystable sheaves on smooth projective surfaces, quadraticity of the Kuranishi family is equivalent to the formality of the DG-Lie algebra of derived endomorphisms.

Findings

Quadraticity of the Kuranishi family implies formality of the DG-Lie algebra.

Formality of the DG-Lie algebra implies quadraticity of the Kuranishi family.

The equivalence holds specifically for polystable sheaves on smooth complex projective surfaces.

Abstract

We study relations between the quadraticity of the Kuranishi family of a coherent sheaf on a complex projective scheme and the formality of the DG-Lie algebra of its derived endomorphisms. In particular, we prove that for a polystable coherent sheaf on a smooth complex projective surface the DG-Lie algebra of derived endomorphisms is formal if and only if the Kuranishi family is quadratic.