Cyclic forms on DG-Lie algebroids and semiregularity

E. Lepri
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引用次数: 3

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 $0$ we define a notion of connection $\nabla \colon \mathbf{R}\Gamma(X,\mathrm{Ker} \rho) \to \mathbf{R}\Gamma (X,\Omega_X^1[-1]\otimes \mathrm{Ker} \rho)$ and construct an $L_\infty$ morphism between DG-Lie algebras $f \colon \mathbf{R}\Gamma(X, \mathrm{Ker} \rho) \rightsquigarrow\mathbf{R}\Gamma(X, \Omega_X^{\leq 1} [2])$ 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 $X$.
DG-Lie代数群上的循环形式及半正则性
给定在特征为$0$的域$\mathbb{K}$上的有限型光滑分离格式$X$上的可传递DG-Lie代数体$(\mathcal{A}, \rho)$,我们定义了连接$\nabla \colon \mathbf{R}\Gamma(X,\mathrm{Ker} \rho) \to \mathbf{R}\Gamma (X,\Omega_X^1[-1]\otimes \mathrm{Ker} \rho)$的概念,并构造了与连接相关的DG-Lie代数体$f \colon \mathbf{R}\Gamma(X, \mathrm{Ker} \rho) \rightsquigarrow\mathbf{R}\Gamma(X, \Omega_X^{\leq 1} [2])$与DG-Lie代数体上的循环形式之间的$L_\infty$态射。通过这种方法,我们得到了改进的Buchweitz-Flenner半正则映射在代数环境中的第一分量的提升,并将其应用于$X$上具有有限局部自由分辨率的相干束的变形理论。另一个应用是(Zariski)主束在$X$上的变形。
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