Complex hypersurfaces in direct products of Riemann surfaces

We study smooth complex hypersurfaces in direct products of closed hyperbolic Riemann surfaces and give a classification in terms of their fundamental groups. This answers a question of Delzant and Gromov on subvarieties of products of Riemann surfaces in the smooth codimension one case. We also answer Delzant and Gromov’s question of which subgroups of a direct product of surface groups are Kähler for two classes: subgroups of direct products of three surface groups, and subgroups arising as the kernel of a homomorphism from the product of surface groups to Z${}^3$. These results will be a consequence of answering the more general question of which subgroups of a direct product of surface groups are the image of a homomorphism from a Kähler group, which is induced by a holomorphic map, for the same two classes. This provides new constraints on Kähler groups.