Vanishing theorems on complex manifolds
type: Item
instance of: scholarly article
title: Vanishing theorems on complex manifolds
zbMATH Open document ID: 0578.32055
author: Bernard Shiffman
author: Andrew J. Sommese
published in: Progress in Mathematics
publication date: 1985-01-01T00:00:00Z
review text: The subject of cohomology vanishing theorems for holomorphic vector bundles on complex manifolds plays an important role in several complex variables and algebraic geometry. The theory began in 1953 with the Kodaira vanishing theorem, but its roots can be traced back to Riemann and Roch for the case of curves and to Picard for surfaces. Although the theory straddles complex analysis and algebraic geometry, this book takes primarily an analytic approach. The authors restrict their attention to compact Kaehler manifolds. This book provides a good survey on this subject. The specific contents of the book are as follows: The first chapter treats complex differential geometry leading up to the Kodaira-Nakano identity. Chapter II deals with the vanishing theorems of Kodaira and Nakano for positive and negative line bundles, the imbedding theorem of Kodaira and the Hodge theory of harmonic forms. In contrast with the Kodaira vanishing theorem, the Nakano vanishing theorem does not hold for semi- positive or semi-negative line bundles. Chapter III presents some generalizations of the Nakano vanishing theorem. Introducing the notion of k-positive and k-negative line bundles, the authors give some generalizations of the Nakano vanishing theorem to these line bundles. They also give another generalization by introducing a powerful inductive technique of slicing by hyperplane sections. Chapter III also explains the relationship between the first Lefschetz theorem and the Nakano vanishing theorem. Chapter IV discusses vanishing theorems for line bundles on a complex projective space and on smooth complete intersections, and a vanishing theorem of Le Potier on Grassmann manifolds. There are two methods of generalizing the concept of positivity and vanishing theorems for line bundles to vector bundles. In chapter V, the concept of ampleness for vector bundles is defined and various vanishing theorems for vector bundles are proved. Chapter VI gives more vanishing theorems for vecto…
Mathematics Subject Classification ID: 32L20
Mathematics Subject Classification ID: 32-02
Mathematics Subject Classification ID: 32Q99
Mathematics Subject Classification ID: 32L10
Mathematics Subject Classification ID: 32J25
Mathematics Subject Classification ID: 14C30
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 3925286
zbMATH Keywords: k-positive line bundle
zbMATH Keywords: cohomology vanishing theorems for holomorphic vector bundles on
zbMATH Keywords: complex manifolds
zbMATH Keywords: compact Kaehler manifolds
zbMATH Keywords: imbedding theorem of Kodaira
zbMATH Keywords: Hodge theory of harmonic forms
zbMATH Keywords: Kodaira vanishing theorem
zbMATH Keywords: Nakano vanishing theorem
zbMATH Keywords: k-negative line bundles
zbMATH Keywords: first Lefschetz theorem
zbMATH Keywords: complex projective space
zbMATH Keywords: vanishing theorem of Le Potier on Grassmann manifolds
zbMATH Keywords: vanishing theorems for vector bundles
zbMATH Keywords: Griffiths
zbMATH Keywords: Ramanujam
zbMATH Keywords: Kawamata
zbMATH Keywords: Viehweg
zbMATH Keywords: Mumford
zbMATH Keywords: Grauert- Riemenschneider
zbMATH Keywords: cohomology vanishing theorems for holomorphic vector bundles on complex manifolds
MaRDI profile type: Publication
Recommended article: Vanishing Theorems for V-Manifolds
Recommended article: Q4263918
Recommended article: Vanishing theorems on complete Kähler manifolds
Recommended article: Vanishing theorems on Hermitian manifolds
Recommended article: Vanishing Theorems for Conformally Compact Manifolds
Recommended article: Vanishing theorems on compact hyper-Kähler manifolds
Recommended article: Vanishing theorems on complete Kähler manifolds and their applications
Recommended article: Vanishing theorems on hypersurfaces in Riemannian manifolds
Recommended article: Vanishing theorems for some classes of Riemann-Cartan manifolds
Recommended article: Vanishing theorems
description: scientific article; zbMATH DE number 3925286