Complex hypersurfaces in direct products of Riemann surfaces

Llosa Isenrich, Claudio

doi:10.2140/agt.2024.24.1467

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Complex hypersurfaces in direct products of Riemann surfaces

Claudio Llosa Isenrich

Algebraic & Geometric Topology 24 (2024) 1467–1486

DOI: 10.2140/agt.2024.24.1467

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 the kernel of a homomorphism from the product of surface groups to $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi>ℤ</mi></mrow><mrow><mn>3</mn></mrow></msup></math>$ . 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.

Keywords

complex hypersurfaces, Riemann surfaces, Kähler groups, subdirect products, surface groups

Mathematical Subject Classification

Primary: 32J27

Secondary: 20F65, 20J05, 32Q15

References

Publication

Received: 29 September 2021

Revised: 23 November 2021

Accepted: 5 November 2022

Published: 28 June 2024

Authors

Claudio Llosa Isenrich
Institute of Algebra and Geometry Karlsruhe Institute of Technology Karlsruhe Germany
https://www.math.kit.edu/iag2/~llosa/

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Volume 24, issue 3 (2024)