Some remarks on vanishing theorems for holomorphic vector bundles
type: Item
instance of: scholarly article
title: Some remarks on vanishing theorems for holomorphic vector bundles
zbMATH Open document ID: 0543.32013
DOI: 10.1007/BF01215497
author: Michael Schneider
published in: Mathematische Zeitschrift
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173435
review text: The main result of this paper is the following precise vanishing theorem: Let X be a projective manifold of dimension n and let E and F be vector bundles of rank r, resp. of rank 1 on X. Suppose that \({\mathcal O}_{{\mathbb{P}}(E)}(1)\) and \(A=(\det E)^{-1}\otimes K_ X^{-1}\otimes F\) are numerically semi-positive line bundles. Assume moreover that either \(c_ 1(A)^ n>0\) or that \(\tilde c_ n(E)>0.\) Then \(H^ q(X,S^{\mu}(E)\otimes F)=0\) for \(q>0,\quad \mu \geq 0.\) Here \(\tilde c{}_ n(E)\) is the component in \(H^{2n}(X,{\mathbb{Z}})\) of \(c(E^*)^{- 1}\). - This generalizes results of Griffiths. Similar improvements can be given for the vanishing theorems of Le Potier and Faltings.
Mathematics Subject Classification ID: 32L20
Mathematics Subject Classification ID: 32L10
Mathematics Subject Classification ID: 14F05
zbMATH DE Number: 3863587
zbMATH Keywords: vanishing theorem
zbMATH Keywords: vector bundles
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3863587