AI 中文总结
本文研究带有$\boldsymbol{T}$作用的仿射辛奇点的T-不动点子概型，构造其导出交并证明对偶定理，进而刻画$G=\boldsymbol{SL}_{n+1}$时仿射Grassmannian切片的不动点映射是否为完全交。
AI 中文摘要
设X是带有$\boldsymbol{T}=T \times \boldsymbol{G}_m$作用的锥形仿射辛奇点，不动点概型$X^T$及映射$X^T \rightarrow X$承载了X几何的大量信息。一般而言，$X^T \rightarrow X$不是完全交，因此我们研究经典轨迹为T-不动点子概型$X^T$的导出交。我们证明X上的辛奇点结构为该导出交的结构层提供了对偶定理，该对偶定理可用于研究此类导出交的结构，尤其能刻画它们的上同调幅度。带有$\boldsymbol{T}$作用的辛奇点的一个重要来源是仿射Grassmannian切片$\boldsymbol{\bar{W}}^{\boldsymbol{\beta}}_{\boldsymbol{\beta}}$，我们特别关注$G=\boldsymbol{SL}_{n+1}$时的这类切片，并用先前发展的理论刻画$(\bar{W}^{\beta}_{\beta})^T \rightarrow \bar{W}^{\beta}_{\beta}$何时为完全交。
英文摘要
For $X$ a conical affine symplectic singularity with $\mathbb{T}=T \times \mathbb{G}_m$-action, the fixed scheme $X^T$ and the map $X^T \rightarrow X$ carry much information about the geometry of $X$. In general, $X^T \rightarrow X$ fails to be a complete intersection. Thus, we study a derived intersection whose classical locus is the $T$-fixed subscheme $X^T$. We show that the structure of the symplectic singularity on $X$ produces a duality theorem for the structure sheaf of the derived intersection. The duality theorem allows us to study the structure of such derived intersections; in particular we describe their cohomological amplitude. An important source of symplectic singularities with $\mathbb{T}$-action are affine Grassmannian slices $\overline{W}^λ_μ$. We pay particular attention to these slices when $G=\mathrm{SL}_{n+1}$, and we use the previously developed theory to characterize when $(\overline{W}^λ_μ)^T \rightarrow \overline{W}^λ_μ$ is a complete intersection.
Comments30 pages, 2 figures
