On ampleness of vector bundles
type: Item
instance of: scholarly article
title: On ampleness of vector bundles
zbMATH Open document ID: 1475.14080
author: Snehajit Misra
author: Nabanita Ray
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2021-09-03T00:00:00Z
review text: The paper under review gives a necessary and sufficient condition for the ampleness of a semistable holomorphic bundle with zero discriminant on a complex projective variety. Recall that a holomorphic bundle \(E\) on a projective complex variety \(X\) is \textit{ample} if the associated tautological bundle \(\mathcal{O}_{\mathbb{P}(E)}(1)\) on the projective bundle \(\mathbb{P}(E)\) is ample, namely if \(\mathcal{O}_{\mathbb{P}(E)}(1)\) supports a smooth hermitian metric with positive curvature. On the other hand, \(E\) is called \textit{Griffiths positive} whenever \(E\) supports on itself a smooth hermitian metric with positive curvature. It is known that if \(E\) is Griffiths positive then \(E\) is ample and a famous conjecture states the converse. Moreover, it is also well-known that ampleness of \(E\) implies ampleness of \(det(E)\) but the converse is not true in general. The main contribution of this paper is to close this circle of statements for semistable bundle with zero discriminant (Theorem 1): in this particular case, let \((E,h)\) be a vector bundle on a projective complex variety with hermetian metric \(h\) such that \(det(E)\) is ample. Then \((E,h)\) is Griffiths positive.
reviewed by: Joan Pons-Llopis
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 14J40
Mathematics Subject Classification ID: 14C17
zbMATH DE Number: 7390659
zbMATH Keywords: ample bundles
zbMATH Keywords: complex projective varieties
MaRDI profile type: Publication
full work available at URL: crmath.222
OpenAlex ID: W3196743452
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DOI: 10.5802/CRMATH.222
description: scientific article; zbMATH DE number 7390659