Virtual localization revisited

TL;DR

This paper extends virtual localization formulas to quasi-smooth derived schemes and stacks, removing previous restrictions and broadening applicability in algebraic geometry.

Contribution

It generalizes the virtual localization formula to derived schemes and stacks, using virtual operations without requiring global resolutions.

Findings

Proves the virtual localization formula for quasi-smooth derived schemes.

Shows the inverse of the Euler class can be described via virtual operations.

Removes the need for global resolution hypotheses in localization.

Abstract

Let $T$ be a split torus acting on an algebraic scheme $X$ with fixed locus $Z$. Edidin and Graham showed that on localized $T$-equivariant Chow groups, (a) push-forward $i_*$ along $i : Z \to X$ is an isomorphism, and (b) when $X$ is smooth the inverse $(i_*)^{-1}$ can be described via Gysin pullback $i^!$ and cap product with $e(N)^{-1}$, the inverse of the Euler class of the normal bundle $N$. In this paper we show that (b) still holds when $X$ is a quasi-smooth derived scheme (or Deligne-Mumford stack), using virtual versions of the operations $i^!$ and $(-)\cap e(N)^{-1}$. As a corollary we prove the virtual localization formula $[X]^{vir} = i_* ([Z]^{vir} \cap e(N^{vir})^{-1})$ of Graber-Pandharipande without global resolution hypotheses and over arbitrary base fields. We include an appendix on fixed loci of group actions on (derived) stacks which should be of independent interest.