On deformations of pairs (manifold, coherent sheaf)

TL;DR

This paper studies how pairs consisting of a smooth projective manifold and a coherent sheaf deform infinitesimally, introducing a controlling differential graded Lie algebra and proving an analog of a classical trace map theorem.

Contribution

It introduces a differential graded Lie algebra framework for deformations of pairs and establishes a Mukai-Artamkin type theorem for the trace map in this context.

Findings

Differential graded Lie algebra controls deformations of pairs

Proves an analog of Mukai-Artamkin Theorem for trace map

Provides a new approach to deformation theory of pairs

Abstract

We analyse infinitesimal deformations of pairs $(X,\mathcal{F})$ with $\mathcal{F}$ a coherent sheaf on a smooth projective manifold $X$ over an algebraic closed field of characteristic $0$. We describe a differential graded Lie algebra controlling the deformation problem, and we prove an analog of a Mukai-Artamkin Theorem about the trace map.