Strong rigidity of irreducible quotients of polydiscs of finite volume
type: Item
instance of: scholarly article
title: Strong rigidity of irreducible quotients of polydiscs of finite volume
zbMATH Open document ID: 0673.58017
DOI: 10.1007/BF01462883
author: Ngaiming Mok
published in: Mathematische Annalen
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164481
review text: Let X be a Kähler manifold and N be an irreducible quotient of the polydisc \(\Delta^ n\), \(n\geq 2\), i.e. a quotient of \(\Delta^ n\) by a properly discontinuous group \(\Gamma\) of holomorphic automorphisms acting without fixed point. The author proved in Math. Ann. 272, 197-216 (1985; Zbl 0604.58021) that for X and N compact and homotopic, there exists a diffeomorphism f: \(X\to N\) such that for the lifting F: \(\tilde X\to \tilde N\) to the universal coverings all projections \(F_ i: \tilde X\to \tilde N\) are holomorphic or anti-holomorphic. The main purpose of the present article is to generalize this to the setting where X is complete with Ricci curvature bounded from below by a constant, N has finite volume and there exists a harmonic map f: \(X\to N\) having finite Dirichlet energy. The main method of this paper follows ideas of \textit{J. Jost} and \textit{S.- T. Yau}, e.g., see Math. Ann. 271, 143-152 (1985; Zbl 0554.32021) and their previous work, and involves studying certain holomorphic foliations. There are natural foliations on N given by projecting polydiscs \(\Delta^{i-1}\times \{a_ i\}\times \Delta^{n-i}\) and every leaf of these is dense. The pull-backs \({\mathcal F}_ i\) to X are foliations which admit a closed leaf unless \(F_ i\) is either holomorphic or anti-holomorphic. Assume \(F_ i\) is neither. Then certain leaves of \({\mathcal F}_ i\) must be identified to obtain an open holomorphic equivalence relation and this is studied via results of \textit{B. Kaup} [Math. Ann. 183, 6-16 (1969; Zbl 0172.105)] which leads eventually to a contradiction.
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 32M05
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 4102123
zbMATH Keywords: irreducible quotient of polydiscs
zbMATH Keywords: finite volume
reviewed by: Bruce Gilligan
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4102123