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Arithmeticity of the monodromy of some Kodaira fibrations

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

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 [email protected]
Bena Tshishiku
Affiliation:
Department of Mathematics, Brown University, 151 Thayer Street, Providence, RI 02908, USA email [email protected]
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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.

References

Arapura, D., Toward the structure of fibered fundamental groups of projective varieties , J. Éc. Polytech. Math. 4 (2017), 595–611; MR 3665609.Google Scholar
Atiyah, M. F., The signature of fibre-bundles , in Global analysis (papers in honor of K. Kodaira) (University of Tokyo Press, Tokyo, 1969), 73–84; MR 0254864.Google Scholar
Ben Simon, G., Burger, M., Hartnick, T., Iozzi, A. and Wienhard, A., On weakly maximal representations of surface groups , J. Differential Geom. 105 (2017), 375–404; MR 3619307.Google Scholar
Catanese, F., Kodaira fibrations and beyond: methods for moduli theory , Jpn. J. Math. 12 (2017), 91–174; MR 3694930.Google Scholar
Chen, L., The number of fiberings of a surface bundle over a surface , Algebr. Geom. Topol. 18 (2018), 2245–2263; MR 3797066.Google Scholar
Deligne, P. and Mostow, G. D., Monodromy of hypergeometric functions and nonlattice integral monodromy , Publ. Math. Inst. Hautes Études Sci. 63 (1986), 5–89; MR 849651.Google Scholar
Ellenberg, J. S., Superstrong approximation for monodromy groups , in Thin groups and superstrong approximation, Mathematical Sciences Research Institute Publications, vol. 61 (Cambridge University Press, Cambridge, 2014), 51–71; MR 3220884.Google Scholar
Farb, B. and Margalit, D., A primer on mapping class groups, Princeton Mathematical Series, vol. 49 (Princeton University Press, Princeton, NJ, 2012); MR 2850125 (2012h:57032).Google Scholar
Griffiths, P. and Schmid, W., Recent developments in Hodge theory: a discussion of techniques and results , in Discrete subgroups of Lie groups and applications to moduli, Int. Colloq., Bombay, 1973 (Oxford University Press, 1975), 31–127; MR 0419850.Google Scholar
Grunewald, F., Larsen, M., Lubotzky, A. and Malestein, J., Arithmetic quotients of the mapping class group , Geom. Funct. Anal. 25 (2015), 1493–1542; MR 3426060.Google Scholar
Isaacs, I. M., Character theory of finite groups, Pure and Applied Mathematics, vol. 69 (Academic Press [Harcourt Brace Jovanovich], New York–London, 1976); MR 0460423.Google Scholar
Jespers, E., Olteanu, G. and del Río, Á., Rational group algebras of finite groups: from idempotents to units of integral group rings , Algebr. Represent. Theory 15 (2012), 359–377; MR 2892512.Google Scholar
Kodaira, K., A certain type of irregular algebraic surfaces , J. Anal. Math. 19 (1967), 207–215; MR 0216521.Google Scholar
Looijenga, E., Prym representations of mapping class groups , Geom. Dedicata 64 (1997), 69–83; MR 1432535.Google Scholar
McMullen, C. T., Braid groups and Hodge theory , Math. Ann. 355 (2013), 893–946; MR 3020148.Google Scholar
Morita, S., Geometry of characteristic classes, Translations of Mathematical Monographs, vol. 199 (American Mathematical Society, Providence, RI, 2001); translated from the 1999 Japanese original, Iwanami Series in Modern Mathematics; MR 1826571 (2002d:57019).Google Scholar
Platonov, V. and Rapinchuk, A., Algebraic groups and number theory, Pure and Applied Mathematics, vol. 139 (Academic Press, Boston, MA, 1994); translated from the 1991 Russian original by Rachel Rowen; MR 1278263 (95b:11039).Google Scholar
Salter, N., Surface bundles over surfaces with arbitrarily many fiberings , Geom. Topol. 19 (2015), 2901–2923; MR 3416116.Google Scholar
Serre, J.-P., Linear representations of finite groups, Graduate Texts in Mathematics, vol. 42 (Springer, New York–Heidelberg, 1977), translated from the second French edition by Leonard L. Scott; MR 0450380.Google Scholar
Tshishiku, B., Characteristic classes of fiberwise branched surface bundles via arithmetic groups , Michigan Math. J. 67 (2018), 31–58; MR 3770852.Google Scholar
Venkataramana, T. N., Image of the Burau representation at dth roots of unity , Ann. of Math. (2) 179 (2014), 1041–1083; MR 3171758.Google Scholar
Witte Morris, D., Introduction to arithmetic groups (Deductive Press, 2015); MR 3307755.Google Scholar