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Remarks on the universal covering of compact Kähler manifolds. (Remarques sur le revêtement universel des variétés kählériennes compactes.) (French) Zbl 0810.32013
For \(X\) a compact Kähler manifold, the author studies relationships between \(\pi_ 1 (X)\), the positivity of \(\Omega^ 1_ X\) and generic compact subvarieties of the universal cover \(\widetilde X\) of \(X\), first considered by M. Gromov [J. Differ. Geom. 3, No. 1, 263-292 (1991; Zbl 0719.53042)]. One gives in the first section a new proof for the fact that a K3 surface is simply connected and, in the second section, it is proved that \(\pi_ 1 (X)\) is finite if \(X\) is simple. In the third section it is shown the existence of a \(\widetilde \Gamma\)-reduction \(\widetilde \gamma : \widetilde X \to \widetilde Z\) for any connected compact Kähler manifold (analogous to the Stein reduction), where \(\widetilde Z\) parametrises essentially the maximal connected compact complex subvarieties of \(\widetilde X\). The \(\Gamma\)-reduction \(\gamma : X \to Z = \widetilde Z/ \Gamma\) \((\Gamma : = \pi_ 1 (X))\) obtained by taking the quotients, gives a new bimeromorphic invariant \(gd(X) : = \dim Z\), named \(\Gamma\)-dimension of \(X\). The main result of the fourth section is the following: If \(\chi ({\mathcal O}_ X) \neq 0\) and if \(gd(X) = n \leq 3\), then \(X\) is of general type (i.e. \(k(X) = n = gd(X))\). In the fifth section a geometric criterion for the finiteness of the fundamental group \(\pi_ 1 (X)\) is given. As a corollary, a simple proof is obtained for the fact that a rationally connected compact Kähler manifold is simply connected.

MSC:
32J27 Compact Kähler manifolds: generalizations, classification
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
14E20 Coverings in algebraic geometry
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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