论文标题
Beilinson-Drinfeld Grassmannian的旗帜版本
A flag version of Beilinson-Drinfeld Grassmannian for surfaces
论文作者
论文摘要
在本文中，我们定义并研究了Belinson-Drinfeld Grassmannian对曲线被光滑的投影表面$ x $替换的情况，并且在与封闭子机的非线性标志相关的基因座上给出了琐碎的数据。为了做到这一点，我们首先为几乎完美的复合物，完美的复合物和托架的模量建立了一些普遍的正式胶合效果。然后，我们构建一个简单的对象$ fl_x $的封闭式封闭式旗的标志$ x $的封闭式尺寸，这与采用旗帜联合的操作自然相关。我们证明这个简单的对象具有2个透度的属性。对于一个仿制复合物代数组$ g $，我们最终定义了beilinson-drinfeld grassmannian的派生的，flag模拟$ gr_x $，在表面上$ x $上的$ g $ - 捆绑包，并表明beilinson-drinfeld grassmannian for Curves for Curves的大多数属性都可以扩展到我们的标志性中。特别是，我们证明了一个分解公式，是规范平坦连接的存在，并在$ fl_x $和$ gr_x $上定义了手性产品。我们还用$ gr_x $绘制循环组和正循环组的旗帜类似物的动作的构建。要修复$ x $上的``大''标志，我们将``外来''衍生的结构与$ x $的$ g $捆绑式的经典堆栈相关联。简要考虑了Flag Grassmannian对其他完美的堆栈（替换$ g $捆绑的堆栈）的类似物，并且也证明了旗帜分解。
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.
