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We study smooth complex hypersurfaces in direct products of closed hyperbolicRiemann surfaces and give a classification in terms of their fundamentalgroups. This answers a question of Delzant and Gromov on subvarieties ofproducts of Riemann surfaces in the smooth codimension one case. We also answerDelzant and Gromov's question of which subgroups of a direct product of surfacegroups are K\"ahler for two classes: subgroups of direct products of threesurface groups; and subgroups arising as kernel of a homomorphism from theproduct of surface groups to $\mathbb{Z}^3$. These results will be aconsequence of answering the more general question of which subgroups of adirect product of surface groups are the image of a homomorphism, which isinduced by a holomorphic map, for the same two classes. This provides newconstraints on K\"ahler groups.

Complex hypersurfaces in direct products of Riemann surfaces