Parabolic sheaves on higher dimensional varieties
type: Item
instance of: scholarly article
title: Parabolic sheaves on higher dimensional varieties
zbMATH Open document ID: 0782.14013
DOI: 10.1007/BF01444711
published in: Mathematische Annalen
publication date: 1993-03-03T00:00:00Z
full work available at URL: https://eudml.org/doc/164953
review text: This paper is devoted to the study of parabolic sheaves on higher dimensional smooth projective varieties. First of all, the author defines the notions of \(\chi\)-semistability and \(\mu\)-semistability for parabolic sheaves. As in the theory of stable sheaves she gets \(\mu\text{-stable} \Rightarrow\chi\text{-stable}\Rightarrow\chi\text{- semistable}\Rightarrow\mu\text{-semistable}\). As main results she obtains the following: In a flat family of parabolic torsionfree sheaves, the \(\mu\)-semistable (resp. \(\mu\)-stable) parabolic sheaves form an open subset. Let \(E\) be a \(\mu\)-semistable (resp. \(\mu\)-stable) (with respect to \(H)\) torsionfree parabolic sheaf on a smooth projective variety of dimension \(\geq 2\). We assume that the flag on \(E\) is \[ {\mathcal F}:F_ 1(E)=E_ D\supset F_ 2(D)\supset\cdots\supset F_{r+1}(D)=0 \] with \(F_ 1(E)/F_ i(D)\) torsionfree for \(i=2,\ldots,r+1\). Then the restriction of \(E\) to a general complete intersection curve \(Y\) of high degree is \(\mu\)- semistable (resp. \(\mu\)-stable) (with respect to \(H_{| Y})\). The converse of this result also holds. Finally the author studies the relation between parabolic sheaves and representations of \(\pi_ 1(X- D)\), \(D\) being an irreducible and smooth divisor.
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14F35
zbMATH DE Number: 126340
zbMATH Keywords: fundamental group
zbMATH Keywords: semistability
zbMATH Keywords: parabolic sheaves
author: Usha N. Bhosle
reviewed by: Rosa M. Miró-Roig
MaRDI profile type: Publication
Wikidata QID: Q61773066
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OpenAlex ID: W1966022497
description: scientific article; zbMATH DE number 126340