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Semiorthogonal decompositions of projective spaces from small quantum cohomology

Vanja Zuliani

Abstract

In a recent article Halpern-Leistner defines the notion of quasi--convergent path in the space of Bridgeland stability conditions. Such a path induces a semiorthogonal decomposition of the derived category. We investigate quasi-convergent paths in the stability manifold of projective spaces and answer positively to two questions posed by Halpern-Leistner. We construct quasi-convergent paths that start from the geometric region of the stability space and whose central charge is given by a fundamental solution of the quantum differential equation. We also construct quasi-convergent paths whose central charges are the quantum cohomology central charges defined by Iritani.

Semiorthogonal decompositions of projective spaces from small quantum cohomology

Abstract

In a recent article Halpern-Leistner defines the notion of quasi--convergent path in the space of Bridgeland stability conditions. Such a path induces a semiorthogonal decomposition of the derived category. We investigate quasi-convergent paths in the stability manifold of projective spaces and answer positively to two questions posed by Halpern-Leistner. We construct quasi-convergent paths that start from the geometric region of the stability space and whose central charge is given by a fundamental solution of the quantum differential equation. We also construct quasi-convergent paths whose central charges are the quantum cohomology central charges defined by Iritani.

Paper Structure

This paper contains 19 sections, 23 theorems, 95 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Quasi-convergent paths
  3. Bridgeland stability conditions
  4. Quasi convergent paths
  5. Semiorthogonal decomposition
  6. Algebraic stability conditions
  7. Exceptional objects
  8. Stability conditions from exceptional collections
  9. Geometric stability conditions
  10. Quantum cohomology, Gamma Conjecture II, and Quantum cohomology central charge
  11. Quantum cohomology, quantum connection and flat sections
  12. Gamma Conjecture II and quantum cohomology central charges
  13. Quantum cohomology of projective spaces
  14. Quasi-convergent paths from quantum cohomology
  15. General results
  16. ...and 4 more sections

Key Result

Lemma 2.7

For any $E\in {\mathcal{P}}$ the categories $\mathrm{D}^E$ are thick triangulated subcategories that gives an SOD where the order of the components of the SOD is the one induced by $<^{\text{inf}}$.

Figures (6)

  • Figure 1: Domain where we estimate the asymptotic behaviour of the of the asymptotically exponential fundamental solution.
  • Figure 2: Bending the integration path and the associated mutation.
  • Figure 3: Plot of the approximated phases, note that on the horizontal axis we have $\frac{1}{r}$.
  • Figure 4: Configuration of the integration paths and the associated objects in the exceptional collection.
  • Figure 5: Configuration of the integration paths and the associated objects in the exceptional collection.
  • ...and 1 more figures

Theorems & Definitions (63)

  • Conjecture 1.2
  • Definition 2.1: 2007_stab_cond_on_triang_catbayer2019short
  • Definition 2.2: Limit semistable object
  • Definition 2.3: leistner2023stability
  • Definition 2.4
  • Definition 2.5: leistner2023stability
  • Definition 2.6: leistner2023stability
  • Lemma 2.7: leistner2023stability
  • Definition 3.1: Exceptional objects
  • Definition 3.2: Macri07StabCurves
  • ...and 53 more