Virtual localization revisited

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 $i_{\ast }$ along $i:Z\to X$ is an isomorphism, and (b) when $X$ is smooth the inverse $(i_{\ast }{)}^{-1}$ 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 $[X{]}^{vir}=i_{\ast }([Z{]}^{vir}\cap e(N^{vir}{)}^{-1})$ 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.

46 pages, new title and improved exposition; material on stacky concentration and cosection localization will reappear elsewhere