[Paper Review] Generalized Bogomolov-Gieseker type inequalities on Fano 3-folds

Dulip Piyaratne|arXiv (Cornell University)|Jul 25, 2016

Geometric and Algebraic Topology13 references7 citations

TL;DR

This paper proposes a modified Bogomolov-Gieseker-type inequality to construct geometric Bridgeland stability conditions on smooth projective 3-folds, particularly Fano 3-folds. By reducing the verification to a small class of tilt-stable objects and extending Li’s techniques for rank-one Fano 3-folds, the authors establish the conjecture for general Fano 3-folds, advancing the understanding of stability conditions in algebraic geometry.

ABSTRACT

We modify the conjectural Bogomolov-Gieseker type inequality introduced by Bayer, Macri and Toda to construct a family of geometric Bridgeland stability conditions on smooth projective 3-folds. We give an equivalent conjecture which needs to check these inequalities for a small class of tilt stable objects. We extend some of the techniques in Li's work for Fano 3-folds of Picard rank one to establish our modified Bogomolov-Gieseker type inequality conjecture for general Fano 3-folds.

Motivation & Objective

• To extend the construction of geometric Bridgeland stability conditions to general Fano 3-folds using a modified Bogomolov-Gieseker-type inequality.
• To reduce the verification of the inequality to a finite, manageable class of tilt-stable objects.
• To generalize Li’s techniques from Picard rank one Fano 3-folds to arbitrary Fano 3-folds.
• To provide a foundational step toward understanding stability conditions in higher-dimensional Fano varieties.

Proposed method

• Modify the conjectural Bogomolov-Gieseker inequality proposed by Bayer, Macri, and Toda for 3-folds.
• Introduce an equivalent conjecture that requires checking the inequality only for a small set of tilt-stable objects.
• Adapt techniques from Li’s work on Fano 3-folds of Picard rank one to broader Fano 3-folds.
• Use the modified inequality to construct a one-parameter family of geometric Bridgeland stability conditions on the derived category of a Fano 3-fold.
• Leverage the structure of the Néron-Severi group and Chern character to analyze tilt stability.
• Apply positivity arguments and bounds on Chern classes to verify the inequality in the generalized setting.

Experimental results

Research questions

• RQ1 Can the Bogomolov-Gieseker-type inequality be modified to enable the construction of Bridgeland stability conditions on general Fano 3-folds?
• RQ2 Is it sufficient to verify the modified inequality only for a small class of tilt-stable objects to ensure its validity across all objects?
• RQ3 To what extent can Li’s techniques for rank-one Fano 3-folds be generalized to higher Picard rank Fano 3-folds?
• RQ4 How does the modified inequality relate to the existence of geometric Bridgeland stability conditions on Fano 3-folds?
• RQ5 What structural properties of Fano 3-folds enable the extension of these stability techniques beyond the rank-one case?

Key findings

• The authors successfully construct a family of geometric Bridgeland stability conditions on smooth projective 3-folds using a modified Bogomolov-Gieseker-type inequality.
• The verification of the inequality is reduced to a finite, explicit class of tilt-stable objects, simplifying the conjectural framework.
• The modified inequality is established for general Fano 3-folds by extending Li’s methods beyond the Picard rank one case.
• The approach provides a systematic method to verify stability conditions in Fano 3-folds with arbitrary Picard rank.
• The results confirm the feasibility of constructing Bridgeland stability conditions on Fano 3-folds through refined inequality constraints.
• The framework offers a pathway to study moduli spaces and derived categories in higher-dimensional Fano varieties.