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Stability conditions on the canonical line bundle of $\mathbb{P}^3$

Tianle Mao

Abstract

We study the space of stability conditions on the total space of the canonical line bundle over the three dimensional projective space. We construct a family of geometric stability conditions and some subset of the boudary of them, which are algebraic. We also use spherical twists to construct some other stability conditions.

Stability conditions on the canonical line bundle of $\mathbb{P}^3$

Abstract

We study the space of stability conditions on the total space of the canonical line bundle over the three dimensional projective space. We construct a family of geometric stability conditions and some subset of the boudary of them, which are algebraic. We also use spherical twists to construct some other stability conditions.
Paper Structure (26 sections, 74 theorems, 260 equations)

This paper contains 26 sections, 74 theorems, 260 equations.

Table of Contents

  1. Introduction
  2. Background and History
  3. Classical Bogomolov-Gieseker inequality
  4. Sketch of construction and Bogomolov-Gieseker type inequality with $\mathrm{ch}_{3}$
  5. Geometric stability conditions on then local $\mathbb{P}^{3}$
  6. Boundary points
  7. Open questions
  8. Plan of the paper
  9. Acknowledgement
  10. Notation
  11. Preliminaries
  12. Geometric stability conditions on the vector bundle supported on the zero section
  13. Basic settings
  14. slope stability
  15. tilt stability
  16. ...and 11 more sections

Key Result

Proposition 1.2

(Propostion classicalBG) If $X$ satisfies Assumption crucialassumptionintro, then for any slope semistable sheaf $E \in \mathrm{Coh}_{0}(X)$, we have the inequality:

Theorems & Definitions (162)

  • Proposition 1.2
  • Conjecture 1.3
  • Conjecture 1.4
  • Conjecture 1.5
  • Theorem 1.6
  • Lemma 1.7
  • Theorem 1.8
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • ...and 152 more