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Complex hypersurfaces in direct products of Riemann surfaces

Claudio Llosa Isenrich

Algebraic & Geometric Topology 24 (2024) 1467–1486

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 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.

complex hypersurfaces, Riemann surfaces, Kähler groups, subdirect products, surface groups
Mathematical Subject Classification
Primary: 32J27
Secondary: 20F65, 20J05, 32Q15
Received: 29 September 2021
Revised: 23 November 2021
Accepted: 5 November 2022
Published: 28 June 2024
Claudio Llosa Isenrich
Institute of Algebra and Geometry
Karlsruhe Institute of Technology

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