On the cohomology of Kaehler groups

TL;DR

This paper explores the cohomology of Kähler groups, demonstrating that certain Betti numbers do not vanish for groups arising from compact Kähler manifolds, and characterizing when lattices in Lie groups are Kähler.

Contribution

It proves that Kähler groups with unbounded complex linear representations have non-vanishing second or fourth Betti numbers, and characterizes Kähler lattices in real simple Lie groups of high rank.

Findings

Non-vanishing second or fourth Betti number for certain Kähler groups.

Cocompact lattices in high-rank Lie groups are Kähler iff of Hermitian type.

Supports the Carlson-Toledo conjecture on Kähler lattices.

Abstract

This is the geometric part of two papers on the cohomology of Kaehler groups. Using non-Abelian Hodge theory we show that if a finitely presented group with an unbounded complex linear morphism is the fundamental group of a compact Kaehler manifold then its second or its fourth Betti number does not vanish. Combined with our first paper this shows that a cocompact lattice in a real simple Lie group G of sufficiently large real rank is Kaehler if and only if G is of Hermitian type (a conjecture of Carlson and Toledo).