# Moscow Mathematical Journal

Volume 16, Issue 4, October–December 2016 pp. 651–658.

Fundamental Group and Pluridifferentials on Compact Kähler Manifolds

**Authors**:
Yohan Brunebarbe (1) and Frédéric Campana (2)

**Author institution:**(1) Ecole Polytechnique Fédérale de Lausanne, Lausanne, Chaire de Géométrie, Bâtiment MA, Station 8, CH 1015 Lausanne, Suisse

(2) Institut Elie Cartan, Université de Lorraine, 64, Boulevard des Aiguilletes, 54506-Vandoeuvre-les-Nancy, France, and Institut Universitaire de France KIAS (Seoul, South Korea)

**Summary: **

A compact Kähler manifold *X* is shown to be simply connected if its `symmetric cotangent algebra' is trivial. Conjecturally, such a manifold should even be rationally connected. The relative version is also shown: a proper surjective connected holomorphic map *f*: *X* → *S* between connected manifolds induces an isomorphism of fundamental groups if its smooth fibres are as above, and if *X* is Kähler.

2010 Math. Subj. Class. 14C30, 14J40, 14H30, 14F35, 32J18, 32J25, 32J27, 32Q30.

**Keywords:**Fundamental group, rationally connected manifolds, symmetric differentials,

*L*

^{2}cohomology.

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