Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Proper moduli spaces of orthosymplectic complexes

Chenjing Bu

TL;DR

This work constructs proper good moduli spaces for moduli stacks of Bridgeland semistable orthosymplectic complexes on a smooth projective variety $X$, proposing them as compactifications of moduli spaces for principal $G$-bundles with $G \in \{O_n, Sp_{2n}\}$. A general propagation result shows that if a stack has a proper good moduli space, then its finite-group fixed loci and mapping stacks from finite groupoids inherit this property, enabling the orthosymplectic case to arise as a $\mathbb{Z}/2$-fixed locus of semistable complexes. For $X$ of holomorphic type like $K3$ or abelian surfaces, the moduli spaces acquire a Poisson structure via a $0$-shifted symplectic form on the derived enhancement, suggesting rich symplectic geometry and potential resolutions in special cases. The approach relies on the Alper-Halpern-Leistner-Heinloth existence framework and connects fixed-locus and mapping-stack theories to construct compactified moduli spaces with enumerative and geometric implications.

Abstract

We apply the formalism of Alper-Halpern-Leistner-Heinloth to construct proper good moduli spaces for moduli stacks of Bridgeland semistable orthosymplectic complexes on a complex smooth projective variety, which we propose as a candidate for compactifying moduli spaces of principal bundles for the orthogonal and symplectic groups. We also prove some results on good moduli spaces of fixed point stacks and mapping stacks from finite groupoids.

Proper moduli spaces of orthosymplectic complexes

TL;DR

This work constructs proper good moduli spaces for moduli stacks of Bridgeland semistable orthosymplectic complexes on a smooth projective variety , proposing them as compactifications of moduli spaces for principal -bundles with . A general propagation result shows that if a stack has a proper good moduli space, then its finite-group fixed loci and mapping stacks from finite groupoids inherit this property, enabling the orthosymplectic case to arise as a -fixed locus of semistable complexes. For of holomorphic type like or abelian surfaces, the moduli spaces acquire a Poisson structure via a -shifted symplectic form on the derived enhancement, suggesting rich symplectic geometry and potential resolutions in special cases. The approach relies on the Alper-Halpern-Leistner-Heinloth existence framework and connects fixed-locus and mapping-stack theories to construct compactified moduli spaces with enumerative and geometric implications.

Abstract

We apply the formalism of Alper-Halpern-Leistner-Heinloth to construct proper good moduli spaces for moduli stacks of Bridgeland semistable orthosymplectic complexes on a complex smooth projective variety, which we propose as a candidate for compactifying moduli spaces of principal bundles for the orthogonal and symplectic groups. We also prove some results on good moduli spaces of fixed point stacks and mapping stacks from finite groupoids.
Paper Structure (4 sections, 4 theorems, 10 equations)

This paper contains 4 sections, 4 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Fixed loci and mapping stacks
  3. Proper good moduli spaces
  4. Orthosymplectic complexes

Key Result

Theorem 2.5

Let $S$ be an algebraic space, and let $\mathcal{X}$ be an algebraic stack over $S$. Let $\Gamma$ be a finite group, and consider the forgetful morphism Then $p$ is representable by algebraic spaces, so ${\mathcal{M}\mspace{-3mu}\mathit{ap}\mspace{1mu}} (\mathrm{B} \Gamma, \mathcal{X})$ is an algebraic stack. Moreover, if $\mathcal{X}$ is quasi-separated, then $p$ is quasi-compact and quasi-separ

Theorems & Definitions (6)

  • Example 2.4
  • Theorem 2.5
  • Theorem 3.1
  • Lemma 3.2
  • Theorem 4.4
  • Remark 4.5