Richard Hain

Duke University, Durham, North Carolina, United States

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Publications (49)15.34 Total impact

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    Richard Hain
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    ABSTRACT: In this paper normal functions (in the sense of Griffiths) are used to solve and refine geometric questions about moduli spaces of curves. The first application is to a problem posed by Eliashberg: compute the class in the cohomology of M_{g,n}^c of the pullback of the zero section of the universal jacobian along the section that takes [C;x_1,...,x_n] to Sum d_j x_j in Jac (C), where d_1 + ... + d_n = 0. The second application is to slope inequalities of the type discovered by Moriwaki. There is also a discussion of height jumping and its relevance to slope inequalilties.
    02/2011;
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    ABSTRACT: We prove that the first complex homology of the Johnson subgroup of the Torelli group $T_g$ is a non-trivial unipotent $T_g$-module for all $g\ge 4$ and give an explicit presentation of it as a $\Sym H_1(T_g,\C)$-module when $g\ge 6$. We do this by proving that, for a finitely generated group $G$ satisfying an assumption close to formality, the triviality of the restricted characteristic variety implies that the first homology of its Johnson kernel $K$ is a nilpotent module over the corresponding Laurent polynomial ring, isomorphic to the infinitesimal Alexander invariant of the associated graded Lie algebra of $G$. In this setup, we also obtain a precise nilpotence test.
    Journal of the European Mathematical Society 01/2011; 16(2014):805-822. · 1.88 Impact Factor
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    Richard Hain
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    ABSTRACT: These are detailed notes on a variant of the non-abelian cohomology developed by Minhyong Kim in arXiv:0409456 (published in Invent. Math.) to study rational points of varieties over number fields. The current variant is used in arXiv:1001.5008 to study rational points of the restriction of the universal curve to the generic point of M_{g,n}.
    09/2010;
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    Richard Hain
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    ABSTRACT: Suppose that g > 2, that n > 0 and that m > 0. In this paper we show that if E is an irreducible smooth variety which dominates a divisor D in M_{g,n}[m], the moduli space of n-pointed, smooth curves of genus g with a level m structure, then the closure of the image of the monodromy representation pi_1(E,e)--> Sp_g(Zhat) has finite index in Sp_g(Zhat). A similar result is proved for codimension 1 families of the universal principally polarized abelian variety of dimension g > 2. Both results are deduced from a general "non-abelian strictness theorem". The first result is used in arXiv:1001.5008 to control the Galois cohomology of the function field of M_{g,n}[m] in degrees 1 and 2.
    06/2010;
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    Richard Hain
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    ABSTRACT: In this paper we prove a version of Grothendieck's section conjecture for the restriction of the universal complete curve over M_{g,n}, g > 4, to the function field k(M_{g,n}) where k is, for example, a number field. In this version, the fundamental group of the closed fiber is replaced by its ell-adic unipotent completion when n > 1.
    Journal of the American Mathematical Society 01/2010; · 3.57 Impact Factor
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    Richard Hain
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    ABSTRACT: These informal notes are an expanded version of lectures on the moduli space of elliptic curves given at Zhejiang University in July, 2008. Their goal is to introduce and motivate basic concepts and constructions (such as orbifolds and stacks) important in the study of moduli spaces of curves and abelian varieties through the example of elliptic curves. The reason for working with elliptic curves is that most constructions are elementary and explicit in this case. All four approaches to moduli spaces of curves -- complex analytic, topological, algebro-geometric, and number theoretic -- are considered. Topics covered reflect my own biases. Very little, if anything, in these notes is original, except perhaps the selection of topics and the point of view.
    01/2009;
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    Richard Hain, Makoto Matsumoto
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    ABSTRACT: Fix a prime number ℓ. In this paper we develop the theory of relative pro-ℓ completion of discrete and profinite groups—a natural generalization of the classical notion of pro-ℓ completion—and show that the pro-ℓ completion of the Torelli group does not inject into the relative pro-ℓ completion of the corresponding mapping class group when the genus is at least 2. (See Theorem 1 below.) As an application, we prove that when g⩾2, the action of the pro-ℓ completion of the Torelli group Tg,1 on the pro-ℓ fundamental group of a pointed genus g surface is not faithful.The choice of a first-order deformation of a maximally degenerate stable curve of genus g determines an action of the absolute Galois group GQ on the relative pro-ℓ completion of the corresponding mapping class group. We prove that for all g all such representations are unramified at all primes ≠ℓ when the first order deformation is suitably chosen. This proof was communicated to us by Mochizuki and Tamagawa.
    Journal of Algebra - J ALGEBRA. 01/2009; 321(11):3335-3374.
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    Richard Hain
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    ABSTRACT: This paper gives an exposition of relative weight filtrations on completions of mapping class groups associated to a stable degeneration of marked genus g curves. These relative weight filtrations have been constructed using Galois theory (with Matsumoto) and Hodge theory (with Pearlstein and Terasoma). It is shown that the level 0 part of the relative weight filtration is an analogue of a parabolic subalgebra of a Kac-Moody Lie algebra. It is shown that all such subalgebras correspond to equivalence classes of pants decompositions of the surface -- two being equivalent if and only if they determine the same handlebody that the reference surface bounds. One application is to show that handlebody subgroups of mapping class groups contain elements arbitrarily far down the lower central series of Torelli groups. (This result was also obtained independently by Jamie Jorgensen.)
    03/2008;
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    Richard Hain, Makoto Matsumoto
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    ABSTRACT: Fix a prime number ell. In this paper we develop the theory of relative pro-ell completion of discrete and profinite groups -- a natural generalization of the classical notion of pro-ell completion -- and show that the pro-ell completion of the Torelli group does not inject into the relative pro-ell completion of the corresponding mapping class group when the genus is at least 3. As an application, we prove that when g > 2, the action of the pro-ell completion of the Torelli group T_{g,1} on the pro-ell fundamental group of a pointed genus g surface is not faithful. The choice of a first-order deformation of a maximally degenerate stable curve of genus g determines an action of the absolute Galois group G_Q on the relative pro-ell completion of the corresponding mapping class group. We prove that for all g all such representations are unramified at all primes \neq ell when the first order deformation is suitably chosen. This proof was communicated to us by Mochizuki and Tamagawa.
    03/2008;
  • Richard M. Hain
    11/2006: pages 75-83;
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    Richard M. Hain, Steven Zucker
    11/2006: pages 92-106;
  • Richard M. Hain, Steven Zucker
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    ABSTRACT: Without Abstract
    11/2006: pages 107-114;
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    Richard Hain
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    ABSTRACT: Torelli space (in genus g) is the moduli space of compact Riemann surfaces of genus g together with a symplectic basis of their first homology group. It is the quotient of the genus g Teichmuller space by the Torelli group T_g and is a model of the classifying space of T_g. It is known that almost all T_g are not finite complexes. The goal of this note is to present a suite of problems designed to probe the infinite topology of Torelli spaces. Some background is given and a few new results are proved.
    09/2005;
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    M. Kim, R.M. Hain
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    ABSTRACT: The Hyodo-Kato theorem relates the De Rham cohomology of a variety over a local field with semi-stable reduction to the log crystalline cohomology of the special fiber. In this paper we prove an analogue for rational homotopy types. In particular, this gives a comparison isomorphism for fundamental groups.
    Kim, M. and Hain, R.M. (2005) The Hyodo-Kato theorem for rational homotopy types. Mathematical Research Letters, 12 (2). pp. 155-169. ISSN 10732780. 01/2005;
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    M. Kim, R.M. Hain
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    ABSTRACT: We give a definition of the crystalline fundamental group of suitable log schemes in positive characteristic using the techniques of rational homotopy theory applied to the De Rham–Witt complex.
    Kim, M. and Hain, R.M. (2004) A De Rham–Witt approach to crystalline rational homotopy theory. Compositio Mathematica, 140 (5). pp. 1245-1276. ISSN 0010437X. 01/2004;
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    Richard Hain, Makoto Matsumoto
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    ABSTRACT: Fix a prime number . We prove a conjecture stated by Ihara, which he attributes to Deligne, about the action of the absolute Galois group on the pro- completion of the fundamental group of the thrice punctured projective line. Similar techniques are also used to prove part of a conjecture of Goncharov, also about the action of the absolute Galois group on the fundamental group of the thrice punctured projective line. The main technical tool is the weighted completion of a profinite group with respect to a reductive representation (and other appropriate data).
    Compositio Mathematica 10/2003; 139(2):119-167. · 1.02 Impact Factor
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    Richard Hain, Makoto Matsumoto
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    ABSTRACT: Suppose that C is a smooth, projective, geometrically connected curve of genus g > 2 defined over a number field K. Suppose that x is a K-rational point of C. Denote the Lie algebra of the unipotent completion (over Q_ell) of the fundamental group of the corresponding complex analytic curve by p(C,x). This is acted on by both G_K (the Galois group of K) and Gamma, the mapping class group of the pointed complex curve. In this paper we show that the algebraic cycle C_x-C_x^- in the jacobian of C controls the size of the image of G_K in Aut p(C,x). More precisely, we give necessary and sufficient conditions, in terms of two Galois cohomology classes determined by this cycle, for the Zariski closure of the image of G_K in Aut p(C,x) to contain the image of the mapping class group. We also prove an equivalent version for the pro-ell fundamental group, an unpointed version, and a Galois analogue of the Harris-Pulte Theorem.
    07/2003;
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    Richard Hain
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    ABSTRACT: This paper is an expanded version of a talk given at the Current Developments in Mathematics Conference last November (2002) on the work of Wilfred Schmid on periods of limits of Hodge structures. The paper begins with an exposition of the theory of limits of Hodge structures with some emphasis on the geometry. It goes on to discuss (briefly) the periods of the limit mixed Hodge structure on the fundamental group (made unipotent) of the projective line minus 3 points and its connections with mixed zeta numbers and mixed Tate motives over Spec Z through the work Deligne, Goncharov and others.
    06/2003;
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    Richard Hain, David Reed
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    ABSTRACT: In this paper we compute the asymptotics of the metric on the line bundle over the moduli space of curves that arises when attempting to compute the archimedean height of the algebraic cycle $C-C^-$ in the jacobian of a smooth projective curve of genus $g$. One way to express the results is to say that the metric extends (more or less) to the line bundle found by Moriwaki that has non-negative degree on every complete curve in $\Mbar_g$ not contained in the boundary.
    Journal of differential geometry 12/2002; · 1.18 Impact Factor
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    Minhyong Kim, Richard M. Hain
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    ABSTRACT: We prove a comparison isomorphism between the De Rham rational homotopy type of a smooth proper log variety defined over a p-adic field and the crystalline rational homotopy type of a semi-stable reduction mod p.
    11/2002;

Publication Stats

583 Citations
15.34 Total Impact Points

Institutions

  • 1994–2009
    • Duke University
      • Department of Mathematics
      Durham, North Carolina, United States
  • 1987–2006
    • University of Washington Seattle
      • Department of Mathematics
      Seattle, WA, United States
    • Institute for Advanced Study
      Princeton Junction, New Jersey, United States
  • 1988
    • Mount Holyoke College
      South Hadley, Massachusetts, United States