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Deformations of locally constant stability conditions and good moduli spaces

Ian Selvaggi

Abstract

We give a structure result on the set of locally constant stability conditions, $\operatorname{Stab}(\mathcal{D}/R)$, defined by Halpern-Leistner-Robotis showing that it has the structure of a complex manifold, in total analogy with Bridgeland's work. As a consequence, we show that the property of having relative mass-hom bounds and the existence of good moduli spaces depends only on the connected components of $\operatorname{Stab}(\mathcal{D}/R)$. Lastly, we observe that the datum of a locally constant stability condition is equivalent to that of a flat family of stability conditions, as described by Bayer et al. in the context of noncommutative algebraic geometry.

Deformations of locally constant stability conditions and good moduli spaces

Abstract

We give a structure result on the set of locally constant stability conditions, , defined by Halpern-Leistner-Robotis showing that it has the structure of a complex manifold, in total analogy with Bridgeland's work. As a consequence, we show that the property of having relative mass-hom bounds and the existence of good moduli spaces depends only on the connected components of . Lastly, we observe that the datum of a locally constant stability condition is equivalent to that of a flat family of stability conditions, as described by Bayer et al. in the context of noncommutative algebraic geometry.

Paper Structure

This paper contains 15 sections, 46 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Moduli of stable objects in a linear category
  3. Linear categories
  4. Presentable linear categories
  5. Smooth and proper categories
  6. Pseudo-coherent and bounded coherent categories
  7. Base change of Ind-Noetherian $t$-structures
  8. Stability conditions
  9. Existence of moduli spaces and consequences
  10. Local deformations of stability
  11. Flat and torsion objects, representable hom spaces
  12. Quot functors
  13. Relation with Harder--Narasimhan structures
  14. The stability manifold
  15. A possible compactification

Key Result

Theorem 1.1

Let $k$ be an excellent ring of characteristic 0 and $\mathcal{A}$ be a cocomplete $k$-linear abelian category that is locally noetherian. Assume that $\mathcal{M}_\mathcal{A}$, the stack parametrizing flat objects in $\mathcal{A}$, is algebraic and locally of finite type over $k$. Then any quasi-co

Theorems & Definitions (118)

  • Theorem 1.1: alper2023existence, Theorem 7.23
  • Theorem A: Theorem \ref{['thmmain']}
  • Theorem B: Corollary \ref{['corcons']}
  • Theorem C: Proposition \ref{['propcomparison']}
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3: PERRY2019877, Lemma 2.7
  • Definition 2.4
  • Lemma 2.5: PERRY2019877, Lemma 4.10
  • Definition 2.6
  • ...and 108 more