Semisimplicity of the group of biholomorphisms of the universal covering of a compact complex manifold with ample canonical bundle. (English) Zbl 0716.32021

Nadel, Alan Michael

Ann. Math. (2) 132, No. 1, 193-211 (1990).

Let M be a connected compact complex manifold with ample canonical bundle and \(\tilde M\) its universal covering. The author studies the group \(Aut(\tilde M)\) of biholomorphic transformations of \(\tilde M\) which is a Lie group since it preserves a Kähler metric. It is proved that the identity component of this group is a real semisimple Lie group without compact factors. If in addition \(\dim M=2\) then exactly one of the following possibilities is valid: \(\tilde M\) is the 2-ball; \(\tilde M\) is the 2-disk; \(Aut(\tilde M)\) acts properly discontinuously on \(\tilde M\) and contains the group of deck transformations as a subgroup of a finite index.

Reviewer: A.L.Onishchik

MSC:

32M05	Complex Lie groups, group actions on complex spaces
53C55	Global differential geometry of Hermitian and Kählerian manifolds
22E46	Semisimple Lie groups and their representations

Keywords:

biholomorphism; ample bundle; Kähler-Einstein metric; canonical bundle; universal covering; semisimple Lie group