DG-李代数胚上的循环型与半正则性
Cyclic forms on DG-Lie algebroids and semiregularity
摘要：
给定特征为$0$的域$\mathbb{K}$上的有限型光滑分离模式$X$上的可迁DG-李代数胚$(\mathcal{A}, ρ)$，定义了联系$\nabla \colon \mathbf{R}Γ(X,\mathrm{Ker} ρ) \to \mathbf{R}Γ(X,Ω_X^1[-1]\otimes \mathrm{Ker} ρ)$的概念，并构造了DG-李代数胚上与联系和循环形式相关联的DG-李代数$f \colon \mathbf{R}Γ(X, \mathrm{Ker} ρ) \rightsquigarrow\mathbf{R}Γ(X, Ω_X^{\leq 1} [2])$之间的$L_\infty$态射.通过这种方式，我们得到了一个提升的第一分量的修改Buchweitz-Flenner半正则映射的代数背景下，它有一个应用程序的变形理论的相干层$X$承认有限的局部自由分解。另一个应用是$X$上的（Zenki）主丛的变形。
英文摘要：
Given a transitive DG-Lie algebroid $(\mathcal{A}, ρ)$ over a smooth separated scheme $X$ of finite type over a field $\mathbb{K}$ of characteristic $0$ we define a notion of connection $\nabla \colon \mathbf{R}Γ(X,\mathrm{Ker} ρ) \to \mathbf{R}Γ(X,Ω_X^1[-1]\otimes \mathrm{Ker} ρ)$ and construct an $L_\infty$ morphism between DG-Lie algebras $f \colon \mathbf{R}Γ(X, \mathrm{Ker} ρ) \rightsquigarrow\mathbf{R}Γ(X, Ω_X^{\leq 1} [2])$ 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 $X$.
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