Derived Enhancements of $T$-fixed subschemes

Abstract

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 $\overline{W}^{\lambda}_{\mu}$. We pay particular attention to these slices when $G=\mathrm{SL}_{n+1}$, and we use the previously developed theory to characterize when $(\overline{W}^{\lambda}_{\mu})^T \rightarrow \overline{W}^{\lambda}_{\mu}$ is a complete intersection.