Aranha, Khan, Latyntsev, Park, Ravi — Virtual Localization Revisited (2025)

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 \((i_*)\textasciicircum{ -1}\) can be described via Gysin pullback \(i\textasciicircum!\) and cap product with \(e(N)\textasciicircum{ -1}\), 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 \(i\textasciicircum!\) and \((-)\cap e(N)\textasciicircum{ -1}\). As a corollary we prove the virtual localization formula \([X]\textasciicircum{ vir}{} = i_* ([Z]\textasciicircum{ vir}{} \cap e(N\textasciicircum{ vir} )\textasciicircum{ -1} )\) 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.