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Stronger Bogomolov--Gieseker type inequality on quintic threefold

Chunkai Xu

TL;DR

This work strengthens the Bogomolov-Gieseker type inequality for slope-semistable sheaves on smooth quintic varieties by combining a refined restriction theorem for tilt-stable objects with Clifford-type bounds on plane quintic curves. The authors derive an explicit, piecewise-linear bound on the normalized second Chern character $\xi_H(F)$ as a function of the slope $\mu_H(F)$, valid for $\mu_H(F)\in[-1,1]$ on quintic threefolds, thereby improving Toda's conjectural bound and implying its prediction. The method also yields a stronger BG-type inequality on quintic surfaces and relies on restricting to hypersurfaces and applying plane curve Clifford bounds to obtain sharp estimates. These results provide new evidence toward the existence of a Gepner-type Bridgeland stability condition on the quintic threefold and extend the analytical toolkit for stability problems in Calabi–Yau geometry.

Abstract

We establish a stronger Bogomolov--Gieseker type inequality for slope-semistable sheaves on the smooth quintic threefold. Our approach combines a refined restriction theorem for tilt-stable objects with explicit Clifford-type bounds for semistable bundles on plane quintic curves. As a consequence, we obtain an explicit piecewise linear inequality on the Chern characters of any slope-semistable sheaf improving upon the classical Bogomolov--Gieseker bound and implying Toda's conjectural inequality. The method also yields a stronger Bogomolov--Gieseker type inequality on smooth quintic surfaces. These results provide new evidence toward the existence of a Bridgeland stability condition of Gepner type on the quintic threefold.

Stronger Bogomolov--Gieseker type inequality on quintic threefold

TL;DR

This work strengthens the Bogomolov-Gieseker type inequality for slope-semistable sheaves on smooth quintic varieties by combining a refined restriction theorem for tilt-stable objects with Clifford-type bounds on plane quintic curves. The authors derive an explicit, piecewise-linear bound on the normalized second Chern character as a function of the slope , valid for on quintic threefolds, thereby improving Toda's conjectural bound and implying its prediction. The method also yields a stronger BG-type inequality on quintic surfaces and relies on restricting to hypersurfaces and applying plane curve Clifford bounds to obtain sharp estimates. These results provide new evidence toward the existence of a Gepner-type Bridgeland stability condition on the quintic threefold and extend the analytical toolkit for stability problems in Calabi–Yau geometry.

Abstract

We establish a stronger Bogomolov--Gieseker type inequality for slope-semistable sheaves on the smooth quintic threefold. Our approach combines a refined restriction theorem for tilt-stable objects with explicit Clifford-type bounds for semistable bundles on plane quintic curves. As a consequence, we obtain an explicit piecewise linear inequality on the Chern characters of any slope-semistable sheaf improving upon the classical Bogomolov--Gieseker bound and implying Toda's conjectural inequality. The method also yields a stronger Bogomolov--Gieseker type inequality on smooth quintic surfaces. These results provide new evidence toward the existence of a Bridgeland stability condition of Gepner type on the quintic threefold.

Paper Structure

This paper contains 9 sections, 22 theorems, 62 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Stability condition: notations and conventions
  4. Useful Lemmas
  5. Clifford type bound on plane quintic curves
  6. Stronger Bogomolov--Gieseker inequality on quintic surface and quintic threefolds
  7. Lemmas: restriction theorem and reduction lemma
  8. Inequality on quintic surfaces
  9. Inequality on quintic threefolds

Key Result

Theorem 1.1

Let $X$ be a smooth projective complex variety and $H$ an ample divisor on $X$. For any torsion-free $H$-slope-semistable sheaf $E$ on $X$, we have where the discriminant is defined by

Figures (3)

  • Figure 1: Clifford type bound on plane quintic curves
  • Figure 2: BG type inequality on quintic surfaces
  • Figure 3: BG type inequality on quintic threefolds

Theorems & Definitions (42)

  • Theorem 1.1: Bogomolov--Gieseker Inequality bogomolov_holomorphic_1979gieseker_theorem_1979
  • Theorem 1.2: Toda2017GepnerPoint
  • Theorem 1.3: See \ref{['MainTheorem']} and \ref{['MainCorollary']}
  • Theorem 1.4: See \ref{['ChInequalityOnSurface']} and \ref{['CorollaryOnSheaves']}
  • Definition 2.1: Slope stability
  • Proposition 2.2: Harder--Narasimhan filtration
  • Definition 2.3: Tilted heart
  • Definition 2.4: Tilt slope and tilt-stability
  • Definition 2.5: $H$-discriminant
  • Theorem 2.6: bayer_bridgeland_2013, piyaratne_moduli_2019
  • ...and 32 more