Complex hypersurfaces in direct products of Riemann surfaces

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 Kähler 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 Kähler groups.

14 pages, V3: Some changes to the exposition, particularly in the introduction. Also added some new references

References in corpus (5)

• Non-finiteness properties of fundamental groups of smooth projective varieties
• The deficiencies of Kaehler groups
• Kähler groups and subdirect products of surface groups
• Cubulable Kähler groups
• Mapping class groups, multiple Kodaira fibrations, and CAT(0) spaces

Cited by in corpus (2)

• Mapping class groups, multiple Kodaira fibrations, and CAT(0) spaces
• Groups with exotic finiteness properties from complex Morse theory