Joint Deformations of Manifolds, Coherent Sheaves and Sections

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).