Stability of homogeneous vector bundles on \({\mathbb{C}}{\mathbb{P}}_ n\)
type: Item
instance of: scholarly article
title: Stability of homogeneous vector bundles on \({\mathbb{C}}{\mathbb{P}}_ n\)
zbMATH Open document ID: 0734.14004
DOI: 10.1007/BF00181215
author: R. F. Rohmfeld
published in: Geometriae Dedicata
publication date: 1991-01-01T00:00:00Z
review text: The author gives a criterion for the stability of homogeneous vector bundles on \({\mathbb{C}}{\mathbb{P}}_ n\). He shows that the stability of algebraic homogeneous vector bundles on \({\mathbb{C}}{\mathbb{P}}_ n\) can be described in terms of homogeneous subbundles. The basic point for this description is the construction of a unique semistable homogeneous subbundle F for every nonstable homogeneous vector bundle E such that \(\mu (F)>\mu (E)\). Here, \(\mu (E):=\) Chern class of E/rank of E, which we call the slope of E. This means that the existence of F contradicts the stability of E. Therefore, we obtain the following criterion: An algebraic homogeneous vector bundle E on \({\mathbb{C}}{\mathbb{P}}_ n\) is stable if and only if for all homogeneous subbundles F of E with \(0<\text{rank}(F)<text{rank}(E)\) we have \(\mu (F)>\mu (E)\).
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14M17
zbMATH DE Number: 4214188
zbMATH Keywords: stability of algebraic homogeneous vector bundles
zbMATH Keywords: slope
zbMATH Keywords: homogeneous subbundles
reviewed by: Masakazu Muro
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00181215
OpenAlex ID: W1979908963
description: scientific article; zbMATH DE number 4214188