Duke Mathematical Journal

Monodromy of codimension 1 subfamilies of universal curves

Richard Hain

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Suppose that g3, that n0, and that 1. The main result is that if E is a smooth variety that dominates a codimension 1 subvariety D of Mg,n[], the moduli space of n-pointed, genus g, smooth, projective curves with a level structure, then the closure of the image of the monodromy representation π1(E,eo)Spg() has finite index in Spg(). A similar result is proved for codimension 1 families of principally polarized abelian varieties.

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Duke Math. J., Volume 161, Number 7 (2012), 1351-1378.

First available in Project Euclid: 4 May 2012

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Zentralblatt MATH identifier

Primary: 14D05: Structure of families (Picard-Lefschetz, monodromy, etc.) 14H15: Families, moduli (analytic) [See also 30F10, 32G15]
Secondary: 14F35: Homotopy theory; fundamental groups [See also 14H30] 14F45: Topological properties


Hain, Richard. Monodromy of codimension 1 subfamilies of universal curves. Duke Math. J. 161 (2012), no. 7, 1351--1378. doi:10.1215/00127094-1593299. https://projecteuclid.org/euclid.dmj/1336142078

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  • [1] N. A’Campo, Tresses, monodromie et le groupe symplectique, Comment. Math. Helv. 54 (1979), 318–327.
  • [2] W. L. Baily, Jr. and A. Borel, Compactification of arithmetic quotients of bounded symmetric domains, Ann. of Math. (2) 84 (1966), 442–528.
  • [3] A. Borel, Stable real cohomology of arithmetic groups, Ann. Sci. École Norm. Sup. (4) 7 (1974), 235–272.
  • [4] N. Bourbaki, Commutative Algebra: Chapters 1–7, reprint of the 1989 English translation, Elem. Math. (Berlin), Springer, Berlin, 1998.
  • [5] P. Deligne, Théorie de Hodge, II, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5–57; III, 44 (1974), 5–77.
  • [6] A. H. Durfee, Neighborhoods of algebraic sets, Trans. Amer. Math. Soc. 276 (1983), 517–530.
  • [7] M. Goresky and R. MacPherson, Stratified Morse Theory, Ergeb. Math. Grenzgeb. (3) 14, Springer, Berlin, 1988.
  • [8] R. Hain, Infinitesimal presentations of the Torelli groups, J. Amer. Math. Soc. 10 (1997), 597–651.
  • [9] R. Hain, The rational cohomology ring of the moduli space of abelian 3-folds, Math. Res. Lett. 9 (2002), 473–491.
  • [10] R. Hain, “Lectures on moduli spaces of elliptic curves” in Transformation Groups and Moduli Spaces of Curves, Adv. Lect. Math. (ALM) 16, International Press, Somerville, Mass., 2010, 95–166.
  • [11] R. Hain, Rational points of universal curves, J. Amer. Math. Soc. 24 (2011), 709–769.
  • [12] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, corrected reprint of the 1978 original, Grad. Stud. Math. 34, Amer. Math. Soc., Providence, 2001.
  • [13] B. Lasell and M. Ramachandran, Observations on harmonic maps and singular varieties, Ann. Sci. École Norm. Sup. (4) 29 (1996), 135–148.
  • [14] C. R. Matthews, L. N. Vaserstein, and B. Weisfeiler, Congruence properties of Zariski-dense subgroups. I, Proc. London Math. Soc. (3) 48 (1984), 514–532.
  • [15] T. Napier and M. Ramachandran, The L2 ∂̅-method, weak Lefschetz theorems, and the topology of Kähler manifolds, J. Amer. Math. Soc. 11 (1998), 375–396.
  • [16] M. V. Nori, Zariski’s conjecture and related problems, Ann. Sci. École Norm. Sup. (4) 16 (1983), 305–344.
  • [17] M. V. Nori, On subgroups of GLn(Fp), Invent. Math. 88 (1987), 257–275.
  • [18] A. Putman, The Picard group of the moduli space of curves with level structures, Duke Math. J. 161 (2012), 623–674.
  • [19] I. Satake, On the compactification of the Siegel space, J. Indian Math. Soc. (N.S.) 20 (1956), 259–281.