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Relative critical loci and quiver moduli

Tristan Bozec, Damien Calaque, Sarah Scherotzke

Abstract

In this paper we identify the cotangent to the derived stack of representations of a quiver $Q$ with the derived moduli stack of modules over the Ginzburg dg-algebra associated with $Q$. More generally, we extend this result to finite type dg-categories, to a relative setting as well, and to deformations of these. It allows us to recover and generalize some results of Yeung, and leads us to the discovery of seemingly new lagrangian subvarieties in the Hilbert scheme of points in the plane.

Relative critical loci and quiver moduli

Abstract

In this paper we identify the cotangent to the derived stack of representations of a quiver with the derived moduli stack of modules over the Ginzburg dg-algebra associated with . More generally, we extend this result to finite type dg-categories, to a relative setting as well, and to deformations of these. It allows us to recover and generalize some results of Yeung, and leads us to the discovery of seemingly new lagrangian subvarieties in the Hilbert scheme of points in the plane.

Paper Structure

This paper contains 45 sections, 37 theorems, 218 equations.

Table of Contents

  1. Introduction
  2. Motivation: new lagrangian subvarieties
  3. A lagrangian subvariety in the commuting variety
  4. New lagrangian subvarieties of the Hilbert scheme of $\mathbb{C}^2$
  5. Recollection on Nakajima quiver varieties
  6. New lagrangian subvarieties
  7. Derived symplectic geometry
  8. Graded mixed complexes and realizations
  9. Graded complexes
  10. Mixed complexes
  11. Graded mixed complexes
  12. De Rham complex and (closed) forms
  13. Shifted symplectic and lagrangian structures
  14. Shifted presymplectic and isotropic structures
  15. Non-degeneracy conditions
  16. ...and 30 more sections

Key Result

Theorem 1

For a finite type dg-category $\StrLen{A}[\mystrlen] \mathrm{A}$, there is an equivalence of exact $(2-n)$-shifted symplectic stacks between the shifted cotangent stack $\mathbf{T}^*[2-n]\mathbf{Perf}_{ \StrLen{A}[\mystrlen] \mathrm{A} }$ and the perfect moduli $\mathbf{Perf}_{\mathcal{G}_n( \StrLen

Theorems & Definitions (124)

  • Theorem : Theorem \ref{['theo: compare']}
  • Theorem : Theorem \ref{['thm:Gn-functoriality']} & Theorem \ref{['theo: comparerel']}
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Theorem 2.5
  • proof
  • ...and 114 more