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Stability conditions and semiorthogonal decompositions I: quasi-convergence

Daniel Halpern-Leistner, Jeffrey Jiang, Antonios-Alexandros Robotis

Abstract

We develop a framework relating semiorthogonal decompositions of a triangulated category $\mathcal{C}$ to paths in its space of stability conditions. We prove that when $\mathcal{C}$ is the homotopy category of a smooth and proper idempotent complete pre-triangulated dg-category, every semiorthogonal decomposition whose factors admit a Bridgeland stability condition can be obtained from our framework.

Stability conditions and semiorthogonal decompositions I: quasi-convergence

Abstract

We develop a framework relating semiorthogonal decompositions of a triangulated category to paths in its space of stability conditions. We prove that when is the homotopy category of a smooth and proper idempotent complete pre-triangulated dg-category, every semiorthogonal decomposition whose factors admit a Bridgeland stability condition can be obtained from our framework.
Paper Structure (17 sections, 43 theorems, 38 equations, 2 figures)

This paper contains 17 sections, 43 theorems, 38 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Semiorthogonal decompositions from paths
  3. Preliminaries on stability conditions
  4. Quasi-convergent paths
  5. Preorders on P
  6. Filtrations of C
  7. Stability conditions on the subquotients
  8. Numerical quasi-convergent paths
  9. Gluing stability conditions
  10. Preliminaries on homological algebra
  11. Gluing constructions revisited
  12. Gluing and quasi-convergence
  13. Many quasi-convergent paths are eventually glued
  14. Quasi-convergent paths recover semiorthogonal decompositions
  15. The case of curves
  16. ...and 2 more sections

Key Result

Theorem 1.2

For a quasi-convergent path $\sigma_\bullet$, the $\cC_{\preceq E}$ are thick triangulated subcategories of $\cC$, and:

Figures (2)

  • Figure 1: To visualize \ref{['T:firsttheorem']}, let $E_1,\ldots, E_n\in \mathcal{P}$ be given such that $\{E_1,\ldots, E_n\}\to \mathcal{P}/{\mathop{\sim}^{\mathrm{i}}}$ is an ordered bijection. Then $\cC = \langle \cC^{E_1},\ldots, \cC^{E_n}\rangle$. Given $F\in \mathcal{P}$ such that $F \sim^{\mathrm{i}} E_j$, one has $\mathcal{C}_{\preceq F} = \langle \mathcal{C}^{E_1},\ldots, \mathcal{C}^{E_{j-1}}, \mathcal{C}_{\preceq F}^{E_j}\rangle$ by \ref{['L:LePreciExtension']}.
  • Figure 2: We schematize the three above theorems. The condition $\sim \:= \mathop{\sim}^{\mathrm{i}}$ means that the relations are equivalent on $\mathcal{P}$ so that the filtration $\{\cC_{\preceq E}\}_{E\in \mathcal{P}/{\sim}}$ of \ref{['T:firsttheorem']} is admissible with corresponding SOD as in the bottom right of the figure.

Theorems & Definitions (102)

  • Remark 1.1
  • Theorem 1.2: \ref{['P:exit_sequence_filtration']}, \ref{['T:prestabilityonquotient']}
  • Theorem 1.3: \ref{['T:stabilityonquotient']}
  • Theorem 1.4: \ref{['T:recoveringsod']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • ...and 92 more