Complex hypersurfaces in direct products of Riemann surfaces

TL;DR

This paper classifies smooth complex hypersurfaces in products of hyperbolic Riemann surfaces and characterizes which subgroups of surface group products are Kähler, answering longstanding questions in complex geometry.

Contribution

It provides a classification of hypersurfaces and subgroup characterizations in products of Riemann surfaces, addressing open questions by Delzant and Gromov.

Findings

Classified smooth complex hypersurfaces in products of Riemann surfaces.

Identified which subgroups of surface group products are Kähler.

Connected subgroup properties to holomorphic maps and Kähler group constraints.

Abstract

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.