Vanishing theorems, linear and quadratic normality
type: Item
instance of: scholarly article
title: Vanishing theorems, linear and quadratic normality
zbMATH Open document ID: 0618.14023
DOI: 10.1007/BF01389243
author: Thomas Peternell
author: Joseph Le Potier
author: Michael Schneider
published in: Inventiones Mathematicae
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143436
review text: Suppose \(X\subseteq {\mathbb{P}}^ n_{{\mathbb{C}}}\) is a closed submanifold of the complex projective space, of dimension \(d>2/3n\). It is conjectured (by R. Hartshorne) that X is the intersection of (n-d) hypersurfaces. As a first step toward this very difficult conjecture one can study the cohomology \(H^ i(X,{\mathcal O}(m))\), and especially we may ask whether it vanishes for \(0<i<d\), as it should. The paper under review contains some results (for \(m=1)\) in this direction, and it points out a mistake in an earlier article by the reviewer [which attempted to show such theorems; see J. Reine Angew. Math. 326, 136-151 (1981; Zbl 0452.14004)].
Mathematics Subject Classification ID: 14M07
Mathematics Subject Classification ID: 14M10
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 32L20
zbMATH DE Number: 4002265
zbMATH Keywords: Hartshorne conjecture
zbMATH Keywords: small codimension
zbMATH Keywords: complete intersection
MaRDI profile type: Publication
cites work: Transplanting Cohomology Classes in Complex-Projective Space
cites work: The positivity of the Chern classes of an ample vector bundle
cites work: Verschwindungssätze und Untermannigfaltigkeiten kleiner Kodimension des komplex-projektiven Raumes.
cites work: Varieties of small codimension in projective space
cites work: Ample subvarieties of algebraic varieties. Notes written in collaboration with C. Musili
cites work: A computer aided approach to codimension 2 subvarities of Pn, n6.
cites work: Direct images of sheaves of differentials and the Atiyah class
cites work: Q4044928
cites work: Annulation de la cohomologie à valeurs dans un fibre vectoriel holomorphe positif de rang quelconque
cites work: Some remarks on vanishing theorems for holomorphic vector bundles
cites work: Submanifolds of Abelian varieties
cites work: Q3943072
reviewed by: Gerd Faltings
description: scientific article; zbMATH DE number 4002265