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The stacky concentration theorem

Dhyan Aranha, Adeel A. Khan, Alexei Latyntsev, Hyeonjun Park, Charanya Ravi

TL;DR

This work establishes a general stacky concentration theorem that extends torus localization from schemes to algebraic stacks. By formulating a master localization criterion in terms of Chern-class localizations of line bundles, the authors prove that BM, Chow, bordism, and G-theory groups localize to a closed substack under suitable conditions, with a robust localization triangle as the technical backbone. The results encompass DM and Artin stacks, global quotients, and actions by general algebraic groups, and they apply to oriented BM theories, including algebraic bordism and G-theory, with a categorified version discussed. Concrete instances include Higgs moduli and vector-bundle cones, illustrating the utility in moduli problems and equivariant computations. Overall, the paper unifies and extends localization principles for stacks, providing tools for explicit computations and broader theoretical reach across algebraic geometry and motivic homotopy theory.

Abstract

We give a sufficient criterion for the Chow or algebraic bordism groups of an algebraic stack, localized at a set of Chern classes of line bundles, to be concentrated in some closed substack. This is a vast generalization of the torus fixed-point localization theorem in equivariant intersection theory, which is the special case of the stack quotient of a scheme $X$ by an action of a torus $T$. Taking on the one hand an algebraic stack in place of $X$, we deduce a generalization of torus localization to algebraic stacks. Taking on the other hand any algebraic group $G$ instead of $T$, we obtain a localization theorem in $G$-equivariant intersection theory.

The stacky concentration theorem

TL;DR

This work establishes a general stacky concentration theorem that extends torus localization from schemes to algebraic stacks. By formulating a master localization criterion in terms of Chern-class localizations of line bundles, the authors prove that BM, Chow, bordism, and G-theory groups localize to a closed substack under suitable conditions, with a robust localization triangle as the technical backbone. The results encompass DM and Artin stacks, global quotients, and actions by general algebraic groups, and they apply to oriented BM theories, including algebraic bordism and G-theory, with a categorified version discussed. Concrete instances include Higgs moduli and vector-bundle cones, illustrating the utility in moduli problems and equivariant computations. Overall, the paper unifies and extends localization principles for stacks, providing tools for explicit computations and broader theoretical reach across algebraic geometry and motivic homotopy theory.

Abstract

We give a sufficient criterion for the Chow or algebraic bordism groups of an algebraic stack, localized at a set of Chern classes of line bundles, to be concentrated in some closed substack. This is a vast generalization of the torus fixed-point localization theorem in equivariant intersection theory, which is the special case of the stack quotient of a scheme by an action of a torus . Taking on the one hand an algebraic stack in place of , we deduce a generalization of torus localization to algebraic stacks. Taking on the other hand any algebraic group instead of , we obtain a localization theorem in -equivariant intersection theory.
Paper Structure (49 sections, 33 theorems, 91 equations)

This paper contains 49 sections, 33 theorems, 91 equations.

Table of Contents

  1. Main results
  2. Examples, variants, and generalizations
  3. Deligne--Mumford stacks
  4. Artin stacks with finite stabilizers
  5. Global quotient stacks
  6. Locally of finite type stacks
  7. Localizing K-theory classes
  8. General algebraic group actions
  9. Oriented Borel--Moore homology theories
  10. Categorification
  11. Proof
  12. Localization formulas
  13. Contents of the paper
  14. Acknowledgments
  15. Preliminaries on stacks
  16. ...and 34 more sections

Key Result

Theorem 1

Let $X$ be a scheme of finite type over a field with an action of a split torus $T$. Let $i : X^T \hookrightarrow X$ denote the inclusion of the $T$-fixed locus. Then the push-forward map on $T$-equivariant Chow groups induces an isomorphism where the localization is at the set of first Chern classes of nontrivial $1$-dimensional representations of $T$.

Theorems & Definitions (74)

  • Theorem 1
  • Theorem 1: Stacky concentration
  • Theorem 2: Torus concentration
  • Corollary 3
  • Corollary 4: Equivariant concentration
  • Remark 1.1
  • Remark 1.2
  • Definition 1.4
  • Remark 1.6
  • Remark 1.7
  • ...and 64 more