A general homological Kleiman–Bertini theorem

Abstract

Let $G$ be a smooth algebraic group acting on a variety $X$. Let $\mathcal{ℱ}$ and $\mathcal{ℰ}$ be coherent sheaves on $X$. We show that if all the higher $\mathcal{T}or$ sheaves of $\mathcal{ℱ}$ against $G$-orbits vanish, then for generic $g\in G$, the sheaf $\mathcal{T}{or}_j^X\left(g\mathcal{ℱ},\mathcal{ℰ}\right)$ vanishes for all $j\ge 1$. This generalizes a result of Miller and Speyer for transitive group actions and a result of Speiser, itself generalizing the classical Kleiman–Bertini theorem, on generic transversality, under a general group action, of smooth subvarieties over an algebraically closed field of characteristic 0.