One-relator Kaehler groups

Abstract

We prove that a one-relator group $G$ is K\"ahler if and only if either $G$ is finite cyclic or $G$ is isomorphic to the fundamental group of a compact orbifold Riemann surface of genus $g > 0$ with at most one cone point of order $n$: $$< a_1\, b_1\, \,...\, a_g\, b_g\, \mid\, (\prod_{i=1}^g [a_i\, b_i])^n>\, .$$