A note on Bogomolov-Gieseker's inequality in positive characteristic
type: Item
instance of: scholarly article
title: A note on Bogomolov-Gieseker's inequality in positive characteristic
zbMATH Open document ID: 0769.14005
DOI: 10.1215/S0012-7094-91-06418-5
author: Atsushi Moriwaki
published in: Duke Mathematical Journal
publication date: 1992-09-27T00:00:00Z
review text: Let \(X\) be a \(d\)-dimensional nonsingular projective variety with an ample line bundle \(H\) and \(E\) a torsion-free sheaf of rank \(r\) on \(X\) that is so-called \(\mu\)-semistable with respect to \(H\). The Bogomolov-Gieseker inequality \((r-1)c_ 1(E)^ 2\leq 2rc_ 2(E)\) holds if \(\dim X=2\) and the characteristic is zero. This is proven in appendix B of the paper by establishing effective boundedness of families of torsionfree sheaves to obtain an effective estimation for the nonnegativity of \(c_ 2\). --- In appendix A, the author proves the inequality for positive characteristic if the \(n\)-th Frobenius pullback of \(E\) is \(\mu\)-semistable for all \(n\). The main part of the paper obtains, for arbitrary \(d\) and \(r\leq 3\), the inequality with on both sides the intersection product with \(H^{d-1}\) being taken. Appendix C is on the preservation of semistability under restriction to an ample curve.
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 14G15
zbMATH DE Number: 58552
zbMATH Keywords: semistable sheaf
zbMATH Keywords: Bogomolov-Gieseker inequality
zbMATH Keywords: positive characteristic
reviewed by: J. H. de Boer
MaRDI profile type: Publication
cites work: The positivity of the Chern classes of an ample vector bundle
cites work: Stable vector bundles on projective spaces in \(\text{char}p>0\)
cites work: P-Ample Bundles and Their Chern Classes
cites work: On a Theorem of Bogomolov on Chern Classes of Stable Bundles
cites work: Stable vector bundles on an algebraic surface
cites work: On boundedness of families of torsion free sheaves
cites work: Semistable sheaves on projective varieties and their restriction to curves
cites work: Semi-stably polarized fiber spaces and Bogomolov-Gieseker's inequality
cites work: Lectures on Curves on an Algebraic Surface. (AM-59)
cites work: Q3882568
cites work: Q3934506
cites work: Unstable vector bundles and linear systems on surfaces in characteristic p
full work available at URL: s0012-7094-91-06418-5
OpenAlex ID: W1571588307
description: scientific article; zbMATH DE number 58552