Cyclic forms on DG-Lie algebroids and semiregularity

Abstract

Given a transitive DG-Lie algebroid $(\mathcal{A},\rho )$ over a smooth separated scheme $X$ of finite type over a field $\mathbb{K}$ of characteristic zero, we define a notion of connection $\nabla :\mathbf{R}\Gamma (X,\mathrm{Ker}\rho )\to \mathbf{R}\Gamma (X,{\Omega }_X^1[-1]\otimes \mathrm{Ker}\rho )$ and construct an $L_{\infty }$-morphism between DG-Lie algebras $f:\mathbf{R}\Gamma (X,\mathrm{Ker}\rho )⇝\mathbf{R}\Gamma (X,{\Omega }_X^{\le 1}[2])$ associated to a connection and to a cyclic form on the DG-Lie algebroid. In this way, we obtain a lifting of the first component of the modified Buchweitz–Flenner semiregularity map in the algebraic context, which has an application to the deformation theory of coherent sheaves on $X$ admitting a finite locally free resolution. Another application is to the deformations of (Zariski) principal bundles on $X$.