Bogomolov-Gieseker Type Inequality on Calabi-Yau and Fano 3-folds

TL;DR

This paper establishes a Bogomolov-Gieseker type inequality for stable sheaves on Calabi-Yau and Fano 3-folds, impacting the understanding of Chern number conditions for stable sheaves.

Contribution

It proves a new inequality for third Chern characters on specific 3-folds, using advanced techniques like spreading-out and tilt-stability, refining existing conjectures.

Findings

Inequality holds for stable sheaves on Calabi-Yau and Fano 3-folds.

Implication that existing Chern number conjectures need modification.

Method combines spreading-out, vanishings, and Langer's estimation techniques.

Abstract

We prove a Bogomolov-Gieseker type inequality for the third Chern characters of stable sheaves on Calabi-Yau 3-folds and a large class of Fano 3-folds with given rank and first and second Chern classes. The proof uses the spreading-out technique, vanishings from the tilt-stability conditions, and Langer's estimation theorem of the global sections of torsion free sheaves. In particular, the result implies that the conjectural sufficient conditions on the Chern numbers for the existence of stable sheaves on a Calabi-Yau 3-fold by Douglas-Reinbacher-Yau needs to be modified.