Toward the structure of fibered fundamental groups of projective varieties

Donu Arapura

doi:10.5802/jep.52

Toward the structure of fibered fundamental groups of projective varieties
[Vers la structure des groupes fondamentaux fibrés des variétés projectives]

Donu Arapura ¹

¹ Department of Mathematics, Purdue University West Lafayette, IN 47907, U.S.A.

Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 595-611.

Résumé
Abstract

Le groupe fondamental d’une variété projective lisse est dit fibré s’il s’envoie surjectivement sur celui d’une courbe de genre $2$ ou plus. Le but de cet article est d’établir des restrictions fortes sur ces groupes, et en particulier sur ceux des surfaces de Kodaira. Dans le cas spécifique d’une surface de Kodaira, ces résultats se présentent sous la forme de restrictions sur la représentation de monodromie dans le ‘mapping class group’. Lorsque la représentation de monodromie se compose de certaines représentations standard, les images sont Zariski denses dans un groupe semi-simple de type hermitien.

The fundamental group of a smooth projective variety is fibered if it maps onto the fundamental group of a smooth curve of genus $2$ or more. The goal of this paper is to establish some strong restrictions on these groups, and in particular on the fundamental groups of Kodaira surfaces. In the specific case of a Kodaira surface, these results are in the form of restrictions on the monodromy representation into the mapping class group. When the monodromy is composed with certain standard representations, the images are Zariski dense in a semisimple group of Hermitian type.

Reçu le : 2016-02-02
Accepté le : 2017-05-14
Publié le : 2017-05-30

MR Zbl

DOI : 10.5802/jep.52

Classification : 14H30
Keywords: Kähler group, Mumford-Tate group
Mot clés : Groupe de Kähler, groupe de Mumford-Tate

Affiliations des auteurs :

Donu Arapura ¹

¹ Department of Mathematics, Purdue University West Lafayette, IN 47907, U.S.A.

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Toward the structure of fibered fundamental groups of projective varieties},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {595--611},
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Donu Arapura. Toward the structure of fibered fundamental groups of projective varieties. Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 595-611. doi : 10.5802/jep.52. https://jep.centre-mersenne.org/articles/10.5802/jep.52/

Bibliographie
Cité par

[ABC + 96] J. Amorós, M. Burger, K. Corlette, D. Kotschick & D. Toledo - Fundamental groups of compact Kähler manifolds, Mathematical Surveys and Monographs, vol. 44, American Mathematical Society, Providence, RI, 1996 | Zbl

[And92] Y. André - “Mumford-Tate groups of mixed Hodge structures and the theorem of the fixed part”, Compositio Math. 82 (1992) no. 1, p. 1-24 | Numdam | MR | Zbl

[Ara11] D. Arapura - “Homomorphisms between Kähler groups”, in Topology of algebraic varieties and singularities, Contemp. Math., vol. 538, American Mathematical Society, Providence, RI, 2011, p. 95-111 | DOI | MR | Zbl

[BB66] W. L. Baily & A. Borel - “Compactification of arithmetic quotients of bounded symmetric domains”, Ann. of Math. (2) 84 (1966), p. 442-528 | DOI | MR | Zbl

[BP02] I. Biswas & K. H. Paranjape - “The Hodge conjecture for general Prym varieties”, J. Algebraic Geom. 11 (2002) no. 1, p. 33-39 | DOI | MR | Zbl

[Cat08] F. Catanese - “Differentiable and deformation type of algebraic surfaces, real and symplectic structures”, in Symplectic 4-manifolds and algebraic surfaces, Lect. Notes in Math., vol. 1938, Springer, Berlin, 2008, p. 55-167 | DOI | MR | Zbl

[EM47] S. Eilenberg & S. MacLane - “Cohomology theory in abstract groups. II. Group extensions with a non-Abelian kernel”, Ann. of Math. (2) 48 (1947), p. 326-341 | DOI | MR | Zbl

[FM12] B. Farb & D. Margalit - A primer on mapping class groups, Princeton Mathematical Series, vol. 49, Princeton University Press, Princeton, NJ, 2012 | MR

[Fox52] R. H. Fox - “On Fenchel’s conjecture about $F$ -groups”, Mat. Tidsskr. B. 1952 (1952), p. 61-65 | MR | Zbl

[GLLM15] F. Grunewald, M. Larsen, A. Lubotzky & J. Malestein - “Arithmetic quotients of the mapping class group”, Geom. Funct. Anal. 25 (2015) no. 5, p. 1493-1542 | DOI | MR | Zbl

[GM88] M. Goresky & R. MacPherson - Stratified Morse theory, Ergeb. Math. Grenzgeb. (3), vol. 14, Springer-Verlag, Berlin, 1988 | MR | Zbl

[Hel78] S. Helgason - Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc., New York-London, 1978 | MR | Zbl

[Kob12] T. Koberda - “Asymptotic linearity of the mapping class group and a homological version of the Nielsen-Thurston classification”, Geom. Dedicata 156 (2012), p. 13-30 | DOI | MR | Zbl

[Kod67] K. Kodaira - “A certain type of irregular algebraic surfaces”, J. Analyse Math. 19 (1967), p. 207-215 | DOI | MR | Zbl

[Loo97] E. Looijenga - “Prym representations of mapping class groups”, Geom. Dedicata 64 (1997) no. 1, p. 69-83 | DOI | MR | Zbl

[MF82] D. Mumford & J. Fogarty - Geometric invariant theory, Ergeb. Math. Grenzgeb. (3), vol. 34, Springer-Verlag, Berlin, 1982 | MR | Zbl

[Mil94] J. S. Milne - “Shimura varieties and motives”, in Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., vol. 55, American Mathematical Society, Providence, RI, 1994, p. 447-523 | MR | Zbl

[Mil05] J. S. Milne - “Introduction to Shimura varieties”, in Harmonic analysis, the trace formula, and Shimura varieties, Clay Math. Proc., vol. 4, American Mathematical Society, Providence, RI, 2005, p. 265-378 | MR | Zbl

[Moo] B. Moonen - “An introduction to Mumford-Tate groups”, preprint available from http://www.math.ru.nl/~bmoonen

[Mum66] D. Mumford - “Families of abelian varieties”, in Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), American Mathematical Society, Providence, RI, 1966, p. 347-351 | DOI | Zbl

[Mum69] D. Mumford - “A note of Shimura’s paper ‘Discontinuous groups and abelian varieties’”, Math. Ann. 181 (1969), p. 345-351 | DOI | MR | Zbl

[Sat65] I. Satake - “Holomorphic imbeddings of symmetric domains into a Siegel space”, Amer. J. Math. 87 (1965), p. 425-461 | DOI | MR | Zbl

[Sch73] W. Schmid - “Variation of Hodge structure: the singularities of the period mapping”, Invent. Math. 22 (1973), p. 211-319 | DOI | MR | Zbl

[Shi67] G. Shimura - “Construction of class fields and zeta functions of algebraic curves”, Ann. of Math. (2) 85 (1967), p. 58-159 | DOI | MR | Zbl

[Tol90] D. Toledo - “Examples of fundamental groups of compact Kähler manifolds”, Bull. London Math. Soc. 22 (1990) no. 4, p. 339-343 | DOI | Zbl

[WK65] J. A. Wolf & A. Korányi - “Generalized Cayley transformations of bounded symmetric domains”, Amer. J. Math. 87 (1965), p. 899-939 | DOI | MR | Zbl

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