On Kahler extensions of abelian groups

We show that any Kahler extension of a finitely generated abelian group by a surface group of genus g at least 2 is virtually a product. Conversely, we prove that any homomorphism of an even rank, finitely generated abelian group into the genus g mapping class group with finite image gives rise to a Kahler extension. The main tools come from surface topology and known restrictions on Kahler groups.