Bogomolov instability of higher rank sheaves on surfaces in characteristic \(p\)
type: Item
instance of: scholarly article
title: Bogomolov instability of higher rank sheaves on surfaces in characteristic \(p\)
zbMATH Open document ID: 0820.14028
DOI: 10.1007/BF01446644
author: Gábor Megyesi
published in: Mathematische Annalen
publication date: 1995-09-25T00:00:00Z
full work available at URL: https://eudml.org/doc/165306
review text: Let \(X\) be a smooth projective surface over an algebraically closed field of characteristic \(p > 0\), and let \({\mathcal E}\) be a locally free sheaf of rank \(r\) on \(X\) with \(\delta ({\mathcal E}) = (r - 1) c^ 2_ 1 ({\mathcal E}) - 2rc_ 2 ({\mathcal E}) > 0\). We prove that then some pull-back of \({\mathcal E}\) by the Frobenius morphism \(F : X \to X\) is Bogomolov unstable, i.e., there exists \(n \in \mathbb{N}\) such that \(F^{n*} {\mathcal E}\) has a destabilising subsheaf \({\mathcal E}_ 1\) which satisfies the following two conditions: \[ {1 \over p^{2n}} \left( c_ 1 ({\mathcal E}_ 1) - {\text{rk}{\mathcal E}_ 1 \over r} c_ 1 (F^{n*} {\mathcal E}) \right)^ 2 \geq {\delta ({\mathcal E}) \over 2r}, \quad \text{and} \quad \left( c_ 1 ({\mathcal E}_ 1) - {\text{rk} {\mathcal E}_ 1 \over r} c_ 1 (F^{n*} {\mathcal E})\right).\;H > 0 \] for any ample divisor \(H\) on \(X\).
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 14F05
zbMATH DE Number: 759282
zbMATH Keywords: Bogomolov instability
zbMATH Keywords: sheaves on surfaces
zbMATH Keywords: characteristic \(p\)
zbMATH Keywords: Frobenius morphism
zbMATH Keywords: ample divisor
MaRDI profile type: Publication
cites work: HOLOMORPHIC TENSORS AND VECTOR BUNDLES ON PROJECTIVE VARIETIES
cites work: A note on Bogomolov-Gieseker's inequality in positive characteristic
cites work: Frobenius pull-back of vector bundles of rank 2 over non-uniruled varieties
cites work: Unstable vector bundles and linear systems on surfaces in characteristic p
OpenAlex ID: W2032271346
description: scientific article; zbMATH DE number 759282