Virtual localization revisited
Dhyan Aranha, Adeel A. Khan, Alexei Latyntsev, Hyeonjun Park, Charanya Ravi
TL;DR
The paper extends equivariant localization from classical Chow theory to a virtual setting by working with quasi-smooth derived schemes and DM stacks under a split torus action. It constructs virtual analogues of the fixed-locus Gysin pullback and Euler class, proving that localization is invertible on localized Chow groups and establishing a virtual localization formula $[X]^{vir} = i_*( [X^{hT}]^{vir} \cap e_T(N^{vir})^{-1})$ that holds without global resolutions and over arbitrary bases. The development relies on a robust framework of homotopy fixed points, deformation-to-the-normal-cone, and a Thom-like homotopy invariance, enabling a purely intrinsic formulation in derived geometry. The results subsume and generalize Graber–Pandharipande’s localization, extend to higher-rank tori, and yield a simple wall-crossing formula, with an appendix on fixed loci of actions on stacks for independent usefulness. This work broadens the applicability of virtual localization to arithmetic and non-archimedean settings and provides tools compatible with obstruction-theoretic formalisms.
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.