曲面的 Beilinson-Drinfeld Grassmannian 的旗版本
A flag version of Beilinson-Drinfeld Grassmannian for surfaces
AI总结：
本文将 Beilinson-Drinfeld Grassmannian 推广至光滑射影曲面，利用闭子概形的非线性旗给出平凡化数据，通过建立模空间的形式粘合结果与构造具 2-Segal 性质的单纯旗对象，定义了导出旗类似物 $Gr_X$，证明了分解公式、平坦联络及手性积等性质。
摘要：
在本文中，我们定义并研究 Beilinson-Drinfeld Grassmannian 的一种推广，即将曲线替换为光滑射影曲面 $X$，且平凡化数据在适当关联于闭子概形的非线性旗的轨迹上给出。为此，我们首先为几乎完美复形、完美复形与 torsor 的模空间建立了一些一般的形式粘合结果。随后，我们构造了一个单纯对象 $Fl_X$，它由光滑射影曲面 $X$ 的闭子概形的旗构成，且自然关联于对旗取并的运算。我们证明该单纯对象具有 2-Segal 性质。对于仿射复代数群 $G$，我们最终定义了曲面 $X$ 上 $G$-丛的 Beilinson-Drinfeld Grassmannian 的导出旗类似物 $Gr_X$，并证明曲线的 Beilinson-Drinfeld Grassmannian 的大多数性质可推广至我们的旗推广中。特别地，我们证明了分解公式、典范平坦联络的存在性，并在 $Fl_X$ 和 $Gr_X$ 上的合适层上定义了手性积。我们还概述了环路群及其正环路群的旗类似物对 $Gr_X$ 的作用的构造。对于 $X$ 上固定的“大”旗，我们在 $X$ 上 $G$-丛的经典栈上关联了“奇异”导出结构。对于其他 Perf-local 栈（替代 $G$-丛的栈）的旗 Grassmannian 类似物，我们也进行了简要讨论，并证明了它们的旗分解。
英文摘要：
In this paper we define and study a generalization of the Belinson-Drinfeld Grassmannian to the case where the curve is replaced by a smooth projective surface $X$, and the trivialization data are given on loci suitably associated to a nonlinear flag of closed subschemes. In order to do this, we first establish some general formal gluing results for moduli of almost perfect complexes, perfect complexes and torsors. We then construct a simplicial object $Fl_X$ of flags of closed subschemes of a smooth projective surface $X$, naturally associated to the operation of taking union of flags. We prove that this simplicial object has the 2-Segal property. For an affine complex algebraic group $G$, we finally define a derived, flag analog $Gr_X$ of the Beilinson-Drinfeld Grassmannian of $G$-bundles on the surface $X$, and show that most of the properties of the Beilinson-Drinfeld Grassmannian for curves can be extended to our flag generalization. In particular, we prove a factorization formula, the existence of a canonical flat connection, and define a chiral product on suitable sheaves on $Fl_X$ and on $Gr_X$. We also sketch the construction of actions of flags analogs of the loop group and of the positive loop group on $Gr_X$. To fixed ``large'' flags on $X$, we associate ``exotic'' derived structures on the classical stack of $G$-bundles on $X$. Analogs of the flag Grassmannian for other Perf-local stacks (replacing the stack of $G$-bundles) are briefly considered, and flag factorization is proved for them, too.
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