Some results on fundamental groups of Kähler manifolds
The present thesis adds two bits to the knowledge about fundamental groups of (quasi) compact Kähler manifolds.
The study of the fundamental groups of algebraic projective manifolds seems to have originated with a question of J. P. Serre. Extending the target class to compact Kähler manifolds could in principle enlarge the class of groups in question, but whether that happens remains an open question to the date.
The problem of characterizing the class [special characters omitted] of fundamental groups of compact Kähler manifolds can be attacked from two sides. On one side, one has the positive results: classes of groups which are in [special characters omitted] include all finite groups (proved by Serre), Abelian free groups of even rank, co-compact lattices in some Lie groups, fundamental groups of genus g curves, etc. Moreover, the class is closed under taking direct products and finite index subgroups.
On another side, there are many restrictions a group must satisfy if it is to be the fundamental group of a compact Kähler manifold: its rank (and the rank of any finite index subgroup) must be even, the Malcev algebra associated to its pro-nilpotent completion should be the quotient of a free [special characters omitted]-Lie algebra by an ideal generated by two-commutators, it cannot surject (with finitely generated kernel) onto a group with finitely many ends, etc.
Chapter 1 of the present thesis shows that the mapping class groups (of a punctured genus g Riemann surface) cannot be fundamental groups of compact Kähler varieties, in case g = 2. We conjecture the same holds for arbitrary genus g ≥ 3.
Actually, a more general conjecture is formulated, and some supporting evidence is presented. It deals with restrictions on normal subgroups of Kähler manifolds: we expect they cannot be free of rank greater than one.
Chapter 2 adds more to the (indirect) restrictions on a quasi-Kähler group π = π1(X). Namely, it is shown that in fairly general conditions on the variety X, a natural Poisson structure exists on the representation variety Rep(π, G), at least in the case where G is the unitary group.