A differential-geometric approach to deformations of pairs $(X,E)$

TL;DR

This paper develops a differential-geometric framework for understanding deformations of pairs (X, E), where X is a complex manifold and E a holomorphic vector bundle, using DGLA and Maurer--Cartan equations to refine classical results.

Contribution

It introduces a differential operator approach to deformation theory of pairs (X, E), providing a chain level refinement and explicit geometric expressions for tangent and obstruction spaces.

Findings

Derived Maurer--Cartan equation for the deformation problem

Expressed tangent and obstruction spaces via Atiyah extension cohomology

Provided examples of unobstructed deformations

Abstract

This article gives an exposition of the deformation theory for pairs $(X, E)$, where $X$ is a compact complex manifold and $E$ is a holomorphic vector bundle over $X$, adapting an analytic viewpoint \`{a} la Kodaira-Spencer. By introducing and exploiting an auxiliary differential operator, we derive the Maurer--Cartan equation and differential graded Lie algebra (DGLA) governing the deformation problem, and express them in terms of differential-geometric notions such as the connection and curvature of $E$, obtaining a chain level refinement of the classical results that the tangent space and obstruction space of the moduli problem are respectively given by the first and second cohomology groups of the Atiyah extension of $E$ over $X$. As an application, we give examples where deformations of pairs are unobstructed.