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A stacky approach to identifying the semistable locus of bundles

Dario Weissmann, Xucheng Zhang

TL;DR

This work develops a stack-theoretic, moduli-space–driven lens for identifying the semistable locus of principal bundles over a curve. By translating stability into the existence of schematic or adequate moduli spaces and leveraging Alper’s framework, the authors show that the semistable locus $\mathscr{B}un_G^{ss}$ is the unique maximal open substack admitting a schematic (or adequate) moduli space in characteristic zero (and generally for adequate moduli spaces in arbitrary characteristic). They further analyze rank 2 vector bundles to prove a sharp maximality result, while constructing higher-rank counterexamples demonstrating that separated moduli spaces alone do not recover the classical semistability. The approach hinges on translating existence criteria into concrete stability notions ($\Theta$-reductivity, S-completeness, local reductivity) and employing central isogenies to reduce to almost simple cases, yielding both negative and positive results about when a given open substack admits a (separated) moduli space. Overall, the paper provides a robust stacky criterion for semistability and illustrates its limits across ranks, offering new perspective on when GIT-type constructions can produce moduli spaces and when they cannot.

Abstract

We show that the semistable locus is the unique maximal open substack of the moduli stack of principal bundles over a curve that admits a schematic moduli space. For rank $2$ vector bundles it coincides with the unique maximal open substack that admits a separated moduli space, but for higher rank there exist other open substacks that admit separated moduli spaces.

A stacky approach to identifying the semistable locus of bundles

TL;DR

This work develops a stack-theoretic, moduli-space–driven lens for identifying the semistable locus of principal bundles over a curve. By translating stability into the existence of schematic or adequate moduli spaces and leveraging Alper’s framework, the authors show that the semistable locus is the unique maximal open substack admitting a schematic (or adequate) moduli space in characteristic zero (and generally for adequate moduli spaces in arbitrary characteristic). They further analyze rank 2 vector bundles to prove a sharp maximality result, while constructing higher-rank counterexamples demonstrating that separated moduli spaces alone do not recover the classical semistability. The approach hinges on translating existence criteria into concrete stability notions (-reductivity, S-completeness, local reductivity) and employing central isogenies to reduce to almost simple cases, yielding both negative and positive results about when a given open substack admits a (separated) moduli space. Overall, the paper provides a robust stacky criterion for semistability and illustrates its limits across ranks, offering new perspective on when GIT-type constructions can produce moduli spaces and when they cannot.

Abstract

We show that the semistable locus is the unique maximal open substack of the moduli stack of principal bundles over a curve that admits a schematic moduli space. For rank vector bundles it coincides with the unique maximal open substack that admits a separated moduli space, but for higher rank there exist other open substacks that admit separated moduli spaces.
Paper Structure (20 sections, 49 theorems, 94 equations, 1 figure)

This paper contains 20 sections, 49 theorems, 94 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Schematic moduli spaces
  3. Adequate stability
  4. Proof of Theorem $\ref{['tba']}$
  5. The case when $G$ is almost simple and simply connected
  6. The general case
  7. The vector bundles case
  8. Applications
  9. Existence criteria for vector bundles
  10. $\Theta$-reductivity
  11. S-completeness
  12. Local reductivity
  13. Rank 2
  14. More on $\Theta$-reductivity
  15. More on S-completeness
  16. ...and 5 more sections

Key Result

Theorem A

Let $C$ be a smooth projective connected curve over an algebraically closed field $k$ of characteristic $0$ and let $G$ be a connected reductive group over $k$. Then the open substack $\mathscr{B}un_G^{ss} \subseteq \mathscr{B}un_G$ of semistable principal $G$-bundles is the unique maximal open subs

Figures (1)

  • Figure 1: The polygon $\lambda$ in the rank-degree plane

Theorems & Definitions (112)

  • Theorem A: (Char. 0 version of Theorem \ref{['tba']})
  • Theorem B: (Char. 0 version of Theorem \ref{['1724']}, thesis-zhang Theorem A (3))
  • Theorem C: (Theorem \ref{['rank-3-main']}, thesis-zhang Theorem B)
  • Remark
  • Theorem 2.1
  • Definition 2.2: (Analogue of MR3237451, Definition 11.1 (b))
  • Theorem 2.3: (Analogue of MR3237451, Theorem 11.14 (ii))
  • proof
  • Lemma 2.4
  • proof
  • ...and 102 more