Joint deformations of manifolds, coherent sheaves and sections
Donatella Iacono, Marco Manetti
TL;DR
The paper develops a DG-Lie algebraic framework for infinitesimal deformations of triples $(X,\mathcal F,\sigma)$, where $X$ is smooth, $\mathcal F$ is coherent, and $\sigma$ is a global section. Central to the construction is the coherent DG-Lie algebroid $\mathcal C^*(X,\mathcal E^*,s)$, built as the mapping cocone of $e_s: \mathcal P^*(X,\mathcal E^*) \to \mathcal E^*$, whose derived global sections $L$ control deformations via Maurer–Cartan theory and gauge equivalence; this control is shown to be independent of the chosen resolution. The descent framework of Hinich and Fiorenza–Iacono–Martinengo is employed to glue local data through Thom–Whitney totalization, yielding global deformation functors $\text{Def}_{(X,\mathcal F,\sigma)}$. The approach specializes to deformations of pairs $(X,Z)$ with $Z$ a divisor, where the deformation theory is governed by the mapping cone $e_\sigma: \mathcal P(X,\mathcal L) \to \mathcal L$, and it recovers known results on locally trivial and unobstructed deformations, including Calabi–Yau cases with $H^1(X,\mathcal L)=0$. Overall, the work provides a robust DG-Lie algebraic mechanism to study deformations of triples and divisor pairs with concrete hypercohomological interpretations.
Abstract
We describe a differential graded Lie algebra controlling infinitesimal deformations of triples $(X,\mathcal{F},σ)$, where $\mathcal{F}$ is a coherent sheaf on a smooth variety $X$ over an algebraically closed field of characteristic 0 and $σ\in H^0(X,\mathcal{F})$. Then, we apply this result to investigate deformations of pairs (variety, divisor).