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A categorical perspective on non-abelian localization

Daniel Halpern-Leistner

TL;DR

The paper develops a comprehensive non-abelian localization framework for algebraic derived stacks with $\Theta$-stratifications, combining a virtual $K$-theoretic localization formula with a categorified theory based on highest weight $K$-homology cycles. It introduces baric structures and baric completions to control derived self-intersections, defines a canonical Euler class, and proves a localization theorem that expresses $K$-theoretic invariants as sums over stratification centers, with a transparent interpretation of Euler factors. The results enable universal wall-crossing formulas and establish finiteness properties for cohomology on moduli stacks, including the stack of pure 1-dimensional sheaves on a surface, thereby enriching both the geometric and categorical understanding of localization in derived settings. Together, these advances provide a robust toolkit for studying moduli problems with $\Theta$-stratifications and for deriving index formulas in derived contexts with potential Verlinde-type consequences.

Abstract

In equivariant geometry, a localization (a.k.a., concentration) theorem is typically interpreted as a relationship between the equivariant geometry of a space with a group action and the geometry of its fixed locus. We take a different perspective, that of non-abelian localization: a localization theorem relates the geometry of an algebraic stack that is equipped with a $Θ$-stratification to the geometry of the centers of this stratification. We establish a ``virtual'' $K$-theoretic non-abelian localization formula, meaning it applies to algebraic derived stacks with perfect cotangent complexes. We also establish a categorical upgrade of this theorem, by introducing a category of ``highest weight $K$-homology cycles'' with respect to the stratification, and relating the category of highest weight cycles on the stack to those on the centers of its $Θ$-stratification. We apply these results to prove a universal wall-crossing formula, and establish a new finiteness theorem for the cohomology of tautological complexes on the stack of one-dimensional sheaves on an algebraic surface.

A categorical perspective on non-abelian localization

TL;DR

The paper develops a comprehensive non-abelian localization framework for algebraic derived stacks with -stratifications, combining a virtual -theoretic localization formula with a categorified theory based on highest weight -homology cycles. It introduces baric structures and baric completions to control derived self-intersections, defines a canonical Euler class, and proves a localization theorem that expresses -theoretic invariants as sums over stratification centers, with a transparent interpretation of Euler factors. The results enable universal wall-crossing formulas and establish finiteness properties for cohomology on moduli stacks, including the stack of pure 1-dimensional sheaves on a surface, thereby enriching both the geometric and categorical understanding of localization in derived settings. Together, these advances provide a robust toolkit for studying moduli problems with -stratifications and for deriving index formulas in derived contexts with potential Verlinde-type consequences.

Abstract

In equivariant geometry, a localization (a.k.a., concentration) theorem is typically interpreted as a relationship between the equivariant geometry of a space with a group action and the geometry of its fixed locus. We take a different perspective, that of non-abelian localization: a localization theorem relates the geometry of an algebraic stack that is equipped with a -stratification to the geometry of the centers of this stratification. We establish a ``virtual'' -theoretic non-abelian localization formula, meaning it applies to algebraic derived stacks with perfect cotangent complexes. We also establish a categorical upgrade of this theorem, by introducing a category of ``highest weight -homology cycles'' with respect to the stratification, and relating the category of highest weight cycles on the stack to those on the centers of its -stratification. We apply these results to prove a universal wall-crossing formula, and establish a new finiteness theorem for the cohomology of tautological complexes on the stack of one-dimensional sheaves on an algebraic surface.

Paper Structure

This paper contains 21 sections, 43 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Categorical analogue of K-homology
  3. Statements of main results
  4. Notation
  5. Author's note
  6. Baric completion of a Theta-stratum
  7. A brief review of Theta-stratifications
  8. The baric structure on a Theta-stratum
  9. A canonical filtration for derived self-intersections
  10. Baric completion in general
  11. Baric completion in our context
  12. The Euler class of a complex
  13. The non-abelian localization theorem
  14. Highest weight cycles
  15. The non-abelian localization theorem
  16. ...and 6 more sections

Key Result

Theorem 1.1

In the context of H:lfp, suppose $\mcX$ is quasi-compact and equipped with a $\Theta$-stratification relative to $\mcB$, $\mcX = \mcS_0 \cup \cdots \cup \mcS_N$. (See D:relative_theta_stratification.) Then $\tot_\ast$ and $\tot^\sharp$ preserve categories of highest weight cycles, and the resulting For any $[E] \in K_0(\Cycles(\mcX/\mcB)^{<\infty})$ we have $[E] = \sum_\alpha (\tot_\alpha)_\ast \

Theorems & Definitions (112)

  • Remark 1
  • Remark 2
  • Example 1
  • Example 2
  • Example 3
  • Remark 3
  • Theorem 1.1: \ref{['T:non-abelian_localization']}
  • Definition 1: Relative $\Theta$-stratum
  • Lemma 1
  • Definition 2: Relative $\Theta$-stratification
  • ...and 102 more