Joint Deformations of Manifolds, Coherent Sheaves and Sections

TL;DR

This paper develops a differential graded Lie algebra framework to study infinitesimal deformations of triples involving a smooth variety, a coherent sheaf, and a section, with applications to deformations of pairs consisting of a variety and a divisor.

Contribution

It introduces a new algebraic structure to control deformations of complex geometric objects and applies it to pairs of varieties and divisors.

Findings

Established a dg Lie algebra controlling deformations of triples (X, F, σ)

Applied the framework to analyze deformations of pairs (variety, divisor)

Provided insights into the deformation theory of complex algebraic structures

Abstract

We describe a differential graded Lie algebra controlling infinitesimal deformations of triples $(X,\mathcal{F},\sigma)$, where $\mathcal{F}$ is a coherent sheaf on a smooth variety $X$ over a field of characteristic 0 and $\sigma\in H^0(X,\mathcal{F})$. Then, we apply this result to investigate deformations of pairs (variety, divisor).