Deformations of morphisms of coherent sheaves

TL;DR

This paper extends descent theorems for Deligne groupoids to dgLas without negative cohomology and applies it to explicitly control infinitesimal deformations of morphisms of coherent sheaves on smooth varieties.

Contribution

It generalizes Hinich's descent theorem to a broader class of dgLas and constructs an explicit dgLa controlling deformations of sheaf morphisms.

Findings

Extended descent theorem to dgLas with no negative cohomology.

Constructed explicit dgLa controlling deformations.

Applied Thom-Whitney totalisation to geometric data.

Abstract

We generalise Hinich's Theorem of descent of Deligne groupoids to the case where the dgLas involved have no negative cohomology. We apply this result to study the infinitesimal deformations of a morphism $\alpha: {\mathcal F} \to {\mathcal G}$ of coherent sheaves, where both the sheaves $ {\mathcal F}$ and $ {\mathcal G}$ and the map $\alpha$ can be deformed, on a smooth variety over a field of characteristic zero. In particular, we provide an explicit dgLa that controls these deformations via the Deligne functor, applying the Thom-Whitney totalisation to a specific semicosimplicial dgLa, constructed from geometrical data.