Remarks on the universal covering of compact Kähler manifolds. (Remarques sur le revêtement universel des variétés kählériennes compactes.) (French) Zbl 0810.32013

Campana, Fredéric

Bull. Soc. Math. Fr. 122, No. 2, 255-284 (1994).

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.

Reviewer: Vasile Brînzănescu (Bucureşti)

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

Keywords:

positivity of \(\Omega^ 1_ X\); first fundamental group; compact Kähler manifold; universal cover

Citations:

Zbl 0719.53042

References:

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