Vanishing theorems on complete Kähler manifolds
type: Item
instance of: scholarly article
title: Vanishing theorems on complete Kähler manifolds
zbMATH Open document ID: 0568.32018
author: Takeo Ohsawa
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1984-01-01T00:00:00Z
review text: The author proves the following \(L^ 2\)-vanishing theorem. Let X be a complete Kähler manifold of dimension n, let (E,h) be a hermitian bundle over X, and let \(\sigma\) be a d-closed semipositive (1,1)-form on X. Assume that the curvature form for h is equal to or greater than \(\sigma\). Then, for any \({\bar \sigma}\)-closed E-valued (n,q)-form f which is square integrable with respect to \(\sigma\), we can find an E- valued (n,q-1)-form g which is square integrable with respect to \(\sigma\) satisfying \({\bar \partial}g=f\). Here \(q\geq 1\). - Then the author applies the above theorem to obtain the following two vanishing theorems: Let X be a compact Kähler manifold, let Y be an analytic space, let \(f: X\to Y\) be a holomorphic map, and let (E,h) be a hermitian bundle over X. Assume that the curvature form h is equal to or greater than the pull- back of a Kähler metric on Y. Then, \(H^ q(Y,f_*{\mathcal O}(K_ X\otimes E))=0\) for \(q\geq 1\). - Here \(K_ X\) denotes the canonical bundle of X and \(f_*{\mathcal O}(K_ X\otimes E)\) denotes the direct image sheaf of \({\mathcal O}(K_ X\otimes E).\) Theorem. Let X be a 1-convex manifold with maximal compact analytic set A, and let \(E\to X\) be a holomorphic vector bundle. Assume the restriction of E to A is Nakano-semipositive. Then \(H^ q(X,0(K_ X\otimes E))=0\) for \(q\geq 1\). - Several interesting applications of the above results are given.
Mathematics Subject Classification ID: 32L20
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 3906869
zbMATH Keywords: vanishing theorem
zbMATH Keywords: complete Kähler manifold
zbMATH Keywords: hermitian bundle
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195181825
OpenAlex ID: W1997368396
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DOI: 10.2977/PRIMS/1195181825
description: scientific article; zbMATH DE number 3906869