Vanishing theorems on complete Kähler manifolds and their applications
type: Item
instance of: scholarly article
title: Vanishing theorems on complete Kähler manifolds and their applications
zbMATH Open document ID: 0963.32010
author: Lei Ni
published in: Journal of Differential Geometry
publication date: 1999-11-23T00:00:00Z
review text: In this paper vanishing theorems for \(L^2\)-sections of the holomorphic vector bundles, \(E\), over complete Kähler manifolds, \(M\), are studied. Also, some applications are studied. In the first part, for example, it is shown that if \(M\) is a non-parabolic Kähler manifold with nonnegative Ricci curvature and \(E\) is a quasi-positive line bundle over \(M\), then the degree of \(E\) has to be infinity. Moreover, motivated by the work of \textit{Y.-T. Siu} and \textit{S.-T. Yau} [Ann. Math., II. Ser. 105, 225-264 (1977; Zbl 0358.32006) and \textit{R. E. Greene} and \textit{H. Wu} [Duke. Math. J. 49, 731-756 (1982; Zbl 0513.53045)] the gap phenomenon is established: Suppose \(M^m\) is a complete non-parabolic Kähler manifold with nonnegative Ricci curvature. If the Ricci curvature is quasi-positive then the total scalar curvature \(\int_MS(x) dv=\infty\). The estimate of the dimension of the space of \(L^2\) holomorphic sections of a line bundle under certain conditions is obtained as a generalization of a result of \textit{N. Mok} [Math. Z. 191, 303-317 (1986; Zbl 0585.32032)]. Huber's theorem is also generalized. The second part of the paper is related to the following conjecture of Yau and Green-Wu. Conjecture. Suppose \(M\) is a complete noncompact Kähler manifold with positive holomorphic bisectional curvature. Then \(M\) is biholomorphic to \(\mathbb{C}^n\). \textit{N. Mok, Y.-T. Siu} and \textit{S.-T. Yau} proved the conjecture under the additional assumptions [Compos. Math 44, 183-218 (1984; Zbl 0531.32007)]. The improved version of the Mok-Siu-Yau's theorem is proved in the paper.
Mathematics Subject Classification ID: 32L20
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 1367998
zbMATH Keywords: holomorphic vector bundle
zbMATH Keywords: semi-positive line bundle
zbMATH Keywords: quasi-positive Ricci curvature
reviewed by: Novica Blažić
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214510047
OpenAlex ID: W1585870256
Wikidata QID: Q115173455
DOI: 10.4310/JDG/1214510047
description: scientific article; zbMATH DE number 1367998