Strong rigidity of irreducible quotients of polydiscs of finite volume
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.
scientific article; zbMATH DE number 4102123