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Arithmeticity of the monodromy of some Kodaira fibrations

Part of: Differential topology Families, fibrations Special aspects of infinite or finite groups Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  26 November 2019

Nick Salter  and
Bena Tshishiku
Nick Salter
Affiliation:
Department of Mathematics, Columbia University, 2990 Broadway, New York, NY 10027, USA email nks@math.columbia.edu
Bena Tshishiku
Affiliation:
Department of Mathematics, Brown University, 151 Thayer Street, Providence, RI 02908, USA email bena_tshishiku@brown.edu
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Abstract

A question of Griffiths–Schmid asks when the monodromy group of an algebraic family of complex varieties is arithmetic. We resolve this in the affirmative for a class of algebraic surfaces known as Atiyah–Kodaira manifolds, which have base and fibers equal to complete algebraic curves. Our methods are topological in nature and involve an analysis of the ‘geometric’ monodromy, valued in the mapping class group of the fiber.

Keywords

surface bundlesmonodromymapping class groupsarithmetic groups

MSC classification

Primary: 14D05: Structure of families (Picard-Lefschetz, monodromy, etc.) 57R22: Topology of vector bundles and fiber bundles
Secondary: 11F06: Structure of modular groups and generalizations; arithmetic groups 20F65: Geometric group theory
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 1 , January 2020 , pp. 114 - 157
Copyright
© The Authors 2019 

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Footnotes

NS is supported by NSF grant DMS-1703181 and BT is supported by NSF grant DMS-1502794.

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