Richard Hain

Duke University, Durham, North Carolina, United States

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Publications (54)21.57 Total impact

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    Donu Arapura, Alexandru Dimca, Richard Hain
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    ABSTRACT: We show that the fundamental groups of normal complex algebraic varieties share many properties of the fundamental groups of smooth varieties. The jump loci of rank one local systems on a normal variety are related to the jump loci of a resolution and of a smoothing of this variety.
    12/2014;
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    Richard Hain
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    ABSTRACT: This paper is an exposition of the completion of a modular group with respect to its inclusion into SL_2(Q) and the connection with the theory of modular forms and variations of mixed Hodge structure over modular curves. Among the goals of this paper are to give a context to Manin's iterated Shimura integrals (iterated integrals of modular forms) and to study relations in the "maximal Eisenstein quotient" of the completion of SL_2(Z). In particular, we use a computation of Terasoma to prove that Pollack's quadratic relations in the derivation Lie algebra of a rank two free Lie algebra lift to this Eisenstein quotient, and are thus motivic. We also construct the normal functions associated to Hecke eigen cusp forms.
    03/2014;
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    Richard Hain
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    ABSTRACT: The universal elliptic KZB equation is the integrable connection on the pro-vector bundle over M_{1,2} whose fiber over the point corresponding to the elliptic curve E and a non-zero point x of E is the unipotent completion of \pi_1(E-{0},x). This was written down independently by Calaque, Enriquez and Etingof (arXiv:math/0702670), and by Levin and Racinet (arXiv:math/0703237). It generalizes the KZ-equation in genus 0. These notes are in four parts. The first two parts provide a detailed exposition of this connection (following Levin-Racinet); the third is a leisurely exploration of the connection in which, for example, we compute the limit mixed Hodge structure on the unipotent fundamental group of the Tate curve minus its identity. In the fourth part we elaborate on ideas of Levin and Racinet and explicitly compute the connection over the moduli space of elliptic curves with a non-zero abelian differential, showing that it is defined over Q.
    09/2013;
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    Richard Hain
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    ABSTRACT: In this paper we prove that finite index subgroups of genus 3 mapping class and Torelli groups that contain the group generated by Dehn twists on bounding simple closed curves are not Kahler. These results are deduced from explicit presentations of the unipotent (aka, Malcev) completion of genus 3 Torelli groups and of the relative completions of genus 3 mapping class groups. The main results follow from the fact that these presentations are not quadratic. To complete the picture, we compute presentations of completed Torelli and mapping class in genera > 3; they are quadratic.
    Journal of Topology 05/2013; · 0.86 Impact Factor
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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. · 1.42 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}.
    Journal of Algebraic Geometry 09/2010; · 1.06 Impact Factor
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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; · 3.06 Impact Factor
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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 06/2009; 321(11):3335-3374. · 0.60 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
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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;
  • Minhyong Kim, Richard M. Hain
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    ABSTRACT: Let A be a complete discrete valuation ring with perfect residue field k and fraction field F. Assume chark=p>0. Endow A with the natural log structure A∖0↪A and let X be a proper smooth connected fine saturated log scheme over A with generic fiber X F and special fiber Y. If A cr (Y) denotes the crystalline rational homotopy type of Y and A DR (X F ) the de Rham rational homotopy type of X F , then the main result of the paper is the isomorphism A DR (X F )≃A cr (Y)⊗F·
    Mathematical Research Letters 01/2005; 12(2). · 0.63 Impact Factor
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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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    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. · 1.04 Impact Factor

Publication Stats

623 Citations
21.57 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