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Tensor stable moduli stacks and refined representations of quivers

Tarig Abdelgadir, Daniel Chan

Abstract

In this paper, we look at the problem of modular realisations of derived equivalences, and more generally, the problem of recovering a Deligne-Mumford stack $\mathbb{X}$ and a bundle $\mathcal{T}$ on it, via some moduli problem (on $\mathbb{X}$ or $A = \operatorname{End}_{\mathbb{X}} \mathcal{T}$). The key issue is, how does one incorporate some of the monoidal structure of $\operatorname{Coh}(\mathbb{X})$ into the moduli problem. To this end, we introduce a new moduli stack, the tensor stable moduli stack which generalises the notion of the Serre-stable moduli stack. We then show how it can be used both for stack recovery and the modular realisation problem for derived equivalences. We also study the moduli of refined representations and how it addresses these problems. Finally, we relate the two approaches when $\mathcal{T}$ is a tilting bundle which is a direct sum of line bundles.

Tensor stable moduli stacks and refined representations of quivers

Abstract

In this paper, we look at the problem of modular realisations of derived equivalences, and more generally, the problem of recovering a Deligne-Mumford stack and a bundle on it, via some moduli problem (on or ). The key issue is, how does one incorporate some of the monoidal structure of into the moduli problem. To this end, we introduce a new moduli stack, the tensor stable moduli stack which generalises the notion of the Serre-stable moduli stack. We then show how it can be used both for stack recovery and the modular realisation problem for derived equivalences. We also study the moduli of refined representations and how it addresses these problems. Finally, we relate the two approaches when is a tilting bundle which is a direct sum of line bundles.

Paper Structure

This paper contains 15 sections, 34 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Background: quasi-projective stacks
  3. Skyscraper sheaves on stacks
  4. Tensor stable moduli stacks
  5. Tautological moduli problem
  6. Tensor stable moduli of representations
  7. Refined representation and their moduli
  8. Some monoidal notation
  9. Definition of refined representations
  10. From tensor stable to refined representations
  11. From refined representations to tensor stable
  12. Global quotient presentation of $\mathbb{M}_\textup{ref}$
  13. $\mathbb{M}_{\text{ref}}^\circ$ and GIT stability
  14. Mori-dream stacks
  15. An example: GL projective spaces

Key Result

Theorem 1.1

Let $\mathbb{X}$ be a separated quasi-projective stack and suppose that $\mathcal{L}_1\oplus \ldots \oplus \mathcal{L}_s$ is faithful. Then $\mathbb{M}^{\mathcal{L}_1,\ldots,\mathcal{L}_s} \simeq \mathbb{X}$.

Theorems & Definitions (71)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Definition 2.5
  • ...and 61 more