A Chern-Weil approach to deformations of pairs and its applications

We revisit the theory of deformations of pairs $(X, E)$, where $X$ is a compact complex manifold and $E$ is a holomorphic vector bundle over $X$, from an analytic viewpoint \`{a} la Kodaira-Spencer. By introducing and exploiting an auxiliary differential operator, we derive the Maurer-Cartan equation and DGLA governing the deformation problem, and express them in terms of differential-geometric notions such as the connection and curvature of $E$, obtaining a Chern-Weil--type 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 $X$. We also investigate circumstances where deformations of pairs are unobstructed using our analytic approach.