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- Title
- Virtual localization revisited
- KIAS Author
- Park, Hyeonjun,Park, Hyeonjun
- Journal
- ADVANCES IN MATHEMATICS, 2025
- Archive
-
- 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 & lowast; along i : Z -> X is an isomorphism, and (b) when X is smooth the inverse (i & lowast;)-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 Xis a quasi-smooth derived scheme (or Deligne-Mumford stack), using virtual versions of the operations i! and (-) boolean AND e(N)-1. As a corollary we prove the virtual localization formula [X]vir= i & lowast;([Z]vir boolean AND e(Nvir)-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. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
