栈的表示簇与迹映射
Representation Varieties of Stacks and Trace Maps
AI总结：
本文定义导出栈表示簇，推广表示簇到完美栈，构造迹映射与移对称辛结构，推广Goldman的经典辛结构。
AI中文摘要：
我们定义了一个导出栈 $\mathscr{Rep}_n(X)$，它推广了从代数 $A$ 到其导出 $GL_n$ 表示簇的赋值 $A\leadsto \operatorname{Rep}_n(A)$（见 arXiv:1112.1449），将定义域从代数 $A$ 推广到特征 0 域上的完美栈 $X$。对于拟射影经典概形 $X$，我们证明 $\mathscr{Rep}_n(X)$ 包含一个子函子 $\operatorname{Quot}^{n,\text{fr}}_{\mathscr{O}_X^n}(X)\subset \mathscr{Rep}_n(X)$，该子函子实际上由一个几乎有限型的导出概形表示。我们进一步构造了 $X$ 与 $\operatorname{Quot}^{n,\text{fr}}_{\mathscr{O}_X^n}(X)$ 上拟凝聚层的导出范畴之间的 Fourier-Mukai 积分变换，该变换在 Hochschild 同调层次上诱导了一个迹映射，推广了 arXiv:1112.1449（对有限表示交换代数）中构造的迹态射。我们证明子函子 $\operatorname{Quot}^{n,\text{fr}}_{\mathscr{O}_X^n}(X)$ 是点 Quot 概形的标架位置的导出增强，并且该导出概形是长度为 $n$ 的凝聚挠层栈上的一个 $GL_n$-主丛。因此，该栈是拟射影概形的导出特征栈的类似物，并且我们证明对于光滑的 Calabi-Yau $X$，它从 arXiv:1812.11913 中在具有适当支撑的完美复形模栈上构造的移对称辛结构（见 arXiv:1111.3209 的意义）继承了一个移对称辛结构。该结构被证明推广了 Goldman 构造的亏格 1 曲面的特征簇上的经典辛结构 \cite{gold}。
英文摘要：
We define a derived stack $\mathscr{Rep}_n(X)$ which generalizes the assignment $A\leadsto \operatorname{Rep}_n(A)$ to an algebra of its derived $GL_n$-representation variety of arXiv:1112.1449 from algebras $A$ to perfect stacks $X$ over characteristic 0 fields. In the case of a quasi-projective classical scheme $X$, we show that $\mathscr{Rep}_n(X)$ admits a subfunctor $\operatorname{Quot}^{n,\text{fr}}_{\mathscr{O}_X^n}(X)\subset \mathscr{Rep}_n(X)$, which is in fact represented by a derived scheme almost of finite type. We further construct a Fourier-Mukai integral transform between the derived categories of quasi-coherent sheaves on $X$ and $\operatorname{Quot}^{n,\text{fr}}_{\mathscr{O}_X^n}(X)$ which induces a trace map at the level of Hochschild homology generalizing the trace morphism constructed in arXiv:1112.1449 (for finitely presented commutative algebras). We show that the subfunctor $\operatorname{Quot}^{n,\text{fr}}_{\mathscr{O}_X^n}(X)$ is a derived enhancement of the framed locus of the Quot scheme of points and that this derived scheme is a $GL_n$-torsor over the stack of coherent length $n$ torsion sheaves. Hence, this stack is an analog of the derived character stack for quasi-projective schemes, and we show that for smooth, Calabi-Yau $X$, it inherits a shifted symplectic structure in the sense of arXiv:1111.3209 from the one constructed in arXiv:1812.11913 on the moduli stack of perfect complexes with proper support. This structure is shown to give a generalization of the classical symplectic structure on character varieties of surfaces of genus 1 constructed by Goldman \cite{gold}.
