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Invariance of the $$\Gamma$$-dimension for certain Kaehlerian families of dimension 3. (Invariance de la $$\Gamma$$-dimension pour certaines familles kählériennes de dimension 3.) (French) Zbl 1197.32009
Let $$X$$ be a complex projective manifold, then a famous conjecture of Shafarevich claims that the universal cover $$\tilde{X}$$ admits a proper morphism onto a Stein space. So far the Shafarevich conjecture is open even for surfaces, but the fundamental work of F. Campana [Bull. Soc. Math. Fr. 122, No. 2, 255–284 (1994; Zbl 0810.32013)] and J. Kollár [Invent. Math. 113, No. 1, 177–215 (1993; Zbl 0819.14006)] shows that for every compact Kähler manifold $$X$$ there exists a meromorphic map $$\gamma: X \dashrightarrow \Gamma(X)$$ such that for every subvariety $$Z \subset X$$ passing through a general point such that $$\pi_1(Z) \rightarrow \pi_1(X)$$ has finite image, the map $$\gamma$$ contracts $$Z$$. The $$\Gamma$$-dimension of the variety $$X$$ is defined as the dimension of the variety $$\Gamma(X)$$. Kollár conjectured that the $$\Gamma$$-dimension is invariant under deformation, this conjecture has been proven by B. de Oliveira, L. Katzarkov and M. Ramachandran [Geom. Funct. Anal. 12, No. 4, 651–668 (2002; Zbl 1169.14305)] if the fundamental group $$\pi_1(X)$$ is residually unipotent.
In general Kollár’s conjecture is wide open, the paper under review studies the case of surfaces and threefolds. While the surface case follows quickly from a result of Yum-Tong Siu [Discrete groups in geometry and analysis, Pap. Hon. G. D. Mostow 60th Birthday, Prog. Math. 67, 124–151 (1987; Zbl 0647.53052)], the threefold case needs much more effort. More precisely the author shows that if $$\mathcal X \rightarrow B$$ is a smooth family of compact Kähler threefolds such that the fibres are not of general type, then the $$\Gamma$$-dimension is locally constant. In the case of a family of threefolds of general type the statement also holds if a conjecture of Campana on the orbifold fundamental group of certain surfaces is true (we refer to the paper for more details on this interesting problem). An important tool in the proof is a detailed study of the natural fibrations attached to the family of threefolds.

##### MSC:
 32Q30 Uniformization of complex manifolds 14J30 $$3$$-folds 32J27 Compact Kähler manifolds: generalizations, classification
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