On the Malcev Completion of Kähler Groups

The study of compact Kähler manifolds made by Hodge and others shows that a Kähler structure imposes very strong conditions on the homotopy type of a compact complex manifold X . In particular, unlike in the case of compact differentiable or closed complex manifolds, not every finitely presented group G is the fundamental group of a compact Kähler manifold. Such groups are called Kähler groups. This note has been inspired by the recent work of F. Johnson and E. Rees ([JR]) and M. Gromov ([G]), showing that free products, and in particular free groups, are not Kähler. It has been our purpose to extend this result and find other restrictions on the presentations of Kähler groups. This is done by translating properties of cup products in H(X) into properties of the group bracket in π1X , an idea that came out of [JR], and also by examining the Albanese map X → Alb(X) after [C]. We describe an algorithm derived from [St] to compute Γ1/Γ2G,Γ2/Γ3G⊗ R from a given presentation of a group G, and use it to give three conditions for the groups to be Kähler: The Lie algebra GrL2G, equivalent to the holonomy algebra, cannot be free (3.3); oneor two-relator Kähler groups either have a torsion abelianized or have a Malcev completion isomorphic to that of a compact Riemann surface group (4.6); nonfibered Kähler groups must satisfy certain lower bounds for the number of their defining relations, equivalently upper bounds for the rank of Γ2/Γ3G (5.7,5.8). In §1 we recall the real Malcev completion G⊗R of a group G, its equivalent Lie algebra LG, and a 2-step nilpotent Lie algebra GrL2G ∼= (Γ1/Γ2G⊗R)⊕(Γ2/Γ3G⊗ R), which is determined by Γ1/Γ3G/Torsion and is equivalent to the cup products H(X) ∧ H(X) → H(X). This algebra is actually equivalent to the holonomy algebra of G (cf. [Ch], [Ko]), and is more convenient for our computations. By [M2],[DGMS], when G is a Kähler group the algebra GrL2G determines the Malcev