## Sur le groupe fondamental d’une variété kählérienne. (On the fundamental group of a Kähler manifold).(French)Zbl 0661.53049

We show that the cohomology $$H^ 1L_ 2$$ of the fundamental group $$\Gamma$$ of a compact Kähler manifold V is induced by a morphism to a Riemann surface. It follows (generalizing [F. E. A. Johnson and E. G. Rees, Bull. Lond. Math. Soc. 19, 463-466 (1987; Zbl 0608.53061)]) that $$\Gamma$$ does not decompose into a non-trivial free product. Besides, if $$\Gamma$$ is hyperbolic and V is aspherical, then $$H^ iL_ 2(\Gamma)=0\quad for\quad i\neq \dim_{{\mathbb{C}}}V.$$

### MSC:

 53C55 Global differential geometry of Hermitian and Kählerian manifolds 57M05 Fundamental group, presentations, free differential calculus

### Keywords:

cohomology; fundamental group; Kähler manifold

### Citations:

Zbl 0618.53050; Zbl 0608.53061