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Complex algebraic stacks and morphisms of intersection complexes

Matthew Huynh

TL;DR

This work extends the Barthel–Brasselet–Fieseler–Gabber–Kaup construction of associated morphisms to finite-type complex algebraic stacks with affine stabilizers, proving the existence of $\\mu^{f}: f^{*}IC_{\\mathcal{Y}} o IC_{\\mathcal{X}}$ (equivalently $\\lambda_f: IC_{\\mathcal{Y}} o f_{*}IC_{\\mathcal{X}}$) for any morphism $f: \\mathcal{X} o \\mathcal{Y}$. The proof leverages a stack version of the Decomposition Theorem (via Sun’s result) and a resolution-based construction to establish the required commutative diagrams, while addressing non-uniqueness of associated morphisms. As an application, the Borel–Moore fundamental class of a closed substack of a Deligne–Mumford stack with affine stabilizers lifts to a class in the intersection cohomology, providing a robust mechanism to lift cycle data to intersection cohomology in the stack context. Overall, the paper broadens intersection-theoretic tools to the realm of algebraic stacks, with implications for singularities and cohomology in moduli problems.

Abstract

We generalize a construction of Barthel-Brasselet-Fieseler-Gabber-Kaup in the setting of complex varieties to the setting of finite type, complex algebraic stacks. Given two such stacks $\mathcal{X},\mathcal{Y}$ with affine stabilizers, and a morphism between them, we construct a morphism from the pullback of the intersection complex of $\mathcal{Y}$ to the intersection complex of $\mathcal{X}$. As an application, we show that the Borel-Moore fundamental class of a closed substack $\mathcal{Z}$ in a Deligne-Mumford stack $\mathcal{X}$ lifts to a class in the intersection cohomology of $\mathcal{X}$.

Complex algebraic stacks and morphisms of intersection complexes

TL;DR

This work extends the Barthel–Brasselet–Fieseler–Gabber–Kaup construction of associated morphisms to finite-type complex algebraic stacks with affine stabilizers, proving the existence of (equivalently ) for any morphism . The proof leverages a stack version of the Decomposition Theorem (via Sun’s result) and a resolution-based construction to establish the required commutative diagrams, while addressing non-uniqueness of associated morphisms. As an application, the Borel–Moore fundamental class of a closed substack of a Deligne–Mumford stack with affine stabilizers lifts to a class in the intersection cohomology, providing a robust mechanism to lift cycle data to intersection cohomology in the stack context. Overall, the paper broadens intersection-theoretic tools to the realm of algebraic stacks, with implications for singularities and cohomology in moduli problems.

Abstract

We generalize a construction of Barthel-Brasselet-Fieseler-Gabber-Kaup in the setting of complex varieties to the setting of finite type, complex algebraic stacks. Given two such stacks with affine stabilizers, and a morphism between them, we construct a morphism from the pullback of the intersection complex of to the intersection complex of . As an application, we show that the Borel-Moore fundamental class of a closed substack in a Deligne-Mumford stack lifts to a class in the intersection cohomology of .
Paper Structure (7 sections, 6 theorems, 14 equations)

This paper contains 7 sections, 6 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. Statement of the main theorem
  4. Acknowledgements
  5. A corollary of the Decomposition Theorem for stacks
  6. Proof of Theorem \ref{['thm:main_theorem']}
  7. Lifts of classes of closed substacks

Key Result

Theorem 1.2

(See § 1.1 for our convention on stacks). Let $\mathcal{X}$ and $\mathcal{Y}$ be two stacks with affine stabilizers, and let $f:\mathcal{X} \to \mathcal{Y}$ be a morphism between them. Then there exists a morphism ${\mu^f: f^{\ast}IC_{\mathcal{Y}} \to IC_{\mathcal{X}}}$ making the following diagram

Theorems & Definitions (18)

  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Proposition 2.2
  • Remark 2.3
  • proof : Proof of Proposition \ref{['prop:cor_of_DT']}
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 8 more