On the second cohomology of K\"ahler groups

TL;DR

This paper investigates Carlson and Toledo's conjecture that infinite K"ahler groups have nontrivial second cohomology, proving it under certain conditions related to complex variations of Hodge structures and exploring related geometric properties.

Contribution

It proves the conjecture for K"ahler groups admitting unbounded reductive rigid linear representations derived from complex variations of Hodge structures, under specific assumptions.

Findings

Confirmed the conjecture under certain conditions on the $ ext{C}$-VHS.

Analyzed geometric properties of period domains associated with these structures.

Connected the existence of specific representations to topological invariants.

Abstract

Carlson and Toledo conjectured that any infinite fundamental group $\Gamma$ of a compact K\"ahler manifold satisfies $H^2(\Gamma,\R)\not =0$. We assume that $\Gamma$ admits an unbounded reductive rigid linear representation. This representation necessarily comes from a complex variation of Hodge structure ($\C$-VHS) on the K\"ahler manifold. We prove the conjecture under some assumption on the $\C$-VHS. We also study some related geometric/topological properties of period domains associated to such $\C$-VHS.