  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 K\"ahler for two classes: subgroups of direct products of three
surface groups; and subgroups arising as kernel of a homomorphism from the
product of surface groups to $\mathbb{Z}^3$. 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, which is
induced by a holomorphic map, for the same two classes. This provides new
constraints on K\"ahler groups.
