arXiv subjects

Richard Hain

Publications and source records attributed to Richard Hain.

At least 19 recordsLinked to original sources

arXiv2024

Periods of Real Biextensions

A real biextension is a real mixed Hodge structure that is an extension of R(0) by a mixed Hodge structure with weights $-1$ and $-2$. A unipotent real biextension over an algebraic manifold is a variation of mixed Hodge structure over it, each of whose fibers is a real biextension and whose weight graded quotients are do not vary. We show that if a unipotent real biextension has non abelian monodromy, then its ``general fiber'' does not split. This result is a tool for investigating the boundary behaviour of normal functions and is applied in arXiv:2408.07809 to study the boundary behaviour of the normal function of the Ceresa cycle.

math.AG↗

arXiv2024

The Rank of the Normal Functions of the Ceresa and Gross--Schoen Cycles

The main result is that when the genus is at least 3, the rank of the normal function function of the Ceresa cycle over the moduli space of curves has maximal rank. This result was proved independently by Z. Gao and S.-W. Zhang (arXiv:2407.01304) by different methods. In genus 3 we show that the Green--Griffiths invariant of this normal function is a Teichmuller modular form of weight (4,0,-1) and use this to show that the rank of the Ceresa normal function is exactly 1 along the hyperelliptic locus.

math.AG↗

arXiv2024

Mapping Class Groups of Simply Connected Kähler Manifolds

This paper has 3 principal goals: (1) to survey what is know about mapping class and Torelli groups of simply connected compact Kaehler manifolds, (2) supplement these results, and (3) present a list of questions and open problems to stimulate future work. Apart from reviewing general background, the paper focuses on the case of hypersurfaces in projective space. We explain how older results of Carlson--Toledo arXiv:alg-geom/9708002 and recent results of Kreck--Su arXiv:2009.08054 imply that the homomorphism from the fundamental group of the moduli space of hypersurfaces in P^4 to the mapping class group of the underlying manifold has a very large kernel (contains a free group of rank 2) and has image of infinite index. This is in contrast to the case of curves, where the homomorphism is an isomorphism.

math.AG↗

arXiv2023

Hecke Actions on Loops and Periods of Iterated Shimura Integrals

In this paper we show that the classical Hecke correspondences T_N, N>0, act on the free abelian groups generated by the conjugacy classes of the modular group SL_2(Z) and the conjugacy classes of its profinite completion. We show that this action induces a dual action on the ring of class functions of a certain relative unipotent completion of the modular group. This ring contains all iterated integrals of modular forms that are constant on conjugacy classes. It possesses a natural mixed Hodge structure and, after tensoring with Q_ell$, a natural action of the absolute Galois group. Each Hecke operator preserves this mixed Hodge structure and commutes with the action of the absolute Galois group. Unlike in the classical case, the algebra generated by these Hecke operators is not commutative. The appendix by Pham Tiep is not included. It can be found at arXiv:2303.02807.

math.NT↗

arXiv2020

Johnson Homomorphisms

This paper surveys work on generalized Johnson homomorphisms and tools for studying them. The goal is to unite several related threads in the literature and to clarify existing results and relationships among them using Hodge theory. We survey the work of Alekseev, Kawazumi, Kuno and Naef on the Goldman--Turaev Lie bialgebra, and the work of various authors on cohomological methods for determining the stable image of generalized Johnson homomorphisms. Various open problems and conjectures are included.

math.GT↗

arXiv2020

Hodge Theory of the Turaev Cobracket and the Kashiwara--Vergne Problem

In this paper we show that, after completing in the $I$-adic topology, the Turaev cobracket on the vector space freely generated by the closed geodesics on a smooth, complex algebraic curve $X$ with an algebraic framing is a morphism of mixed Hodge structure. We combine this with results of a previous paper (arXiv:1710.06053) on the Goldman bracket to construct torsors of solutions of the Kashiwara--Vergne problem in all genera. The solutions so constructed form a torsor under a prounipotent group that depends only on the topology of the framed surface. We give a partial presentation of these groups. Along the way, we give a homological description of the Turaev cobracket.

math.QA↗

arXiv2020

Notes on the Universal Elliptic KZB Equation

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 π_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.

math.AG↗

arXiv2019

Hodge theory of the Goldman bracket

In this paper we show that, after completion in the I-adic topology, the Goldman bracket on the space spanned by homotopy classes of loops on a smooth, complex algebraic curve is a morphism of mixed Hodge structure. We prove similar statements for the natural action (defined by Kawazumi and Kuno) of the loops in X on paths from one "boundary component" to another. These results are used to construct torsors of isomorphisms of the the completed Goldman Lie algebra with the completion of its associated graded Lie algebra. Such splittings give torsors of partial solutions to the Kashiwara--Vergne problem (arXiv:1611.05581) in all genera. Compatibility of the cobracket with Hodge theory is established in arXiv:1807.09209.

math.QA↗

arXiv2017

Universal Mixed Elliptic Motives

In this paper we construct a Q-linear tannakian category MEM_1 of universal mixed elliptic motives over the moduli space M_{1,1} of elliptic curves. It contains MTM, the category of mixed Tate motives unramified over the integers. Each object of MEM_1 is an object of MTM endowed with an action of SL_2(Z) that is compatible with its structure. Universal mixed elliptic motives can be thought of as motivic local systems over M_{1,1} whose fiber over the tangential base point d/dq at the cusp is a mixed Tate motive. The basic structure of the tannakian fundamental group of MEM is determined and the lowest order terms of all relations are found (using computations of Francis Brown), including the arithmetic relations, which describe the "infinitesimal Galois action". We use the presentation to give a new and more conceptual proof of the Ihara-Takao congruences.

math.AG↗

arXiv2017

Algebraic de Rham theory for weakly holomorphic modular forms of level one

We establish an Eichler-Shimura isomorphism for weakly modular forms of level one. We do this by relating weakly modular forms with rational Fourier coefficients to the algebraic de Rham cohomology of the modular curve with twisted coefficients. This leads to formulae for the periods and quasi-periods of modular forms.

math.NT↗

arXiv2016

Deligne-Beilinson Cohomology of Affine Groups

The goal of this paper is to develop the theory of Deligne-Beilinson cohomology of affine groups with a mixed Hodge structure. The motivation comes from Hodge theory and the study of motives, where such groups appear. Several of Francis Brown's period computations (arXiv:1407.5167) are interpreted as elements of the DB cohomology of the relative unipotent completion of $SL_2(Z)$ and their cup products. The results in this paper are used in arXiv:1403.6443 where they are used to prove that Pollack's quadratic relations are motivic.

math.AG↗

arXiv2015

The Hodge-de~Rham Theory of Modular Groups

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.

math.AG↗

arXiv2014

On the fundamental groups of normal varieties

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.

math.AG↗

arXiv2014

Genus 3 Mapping Class Groups are not Kahler

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. We also show that groups commensurable with hyperelliptic mapping class groups and mapping class groups in genera < 3 are not Kahler.

math.AG↗

arXiv2014

Lectures on Moduli Spaces of Elliptic Curves

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.

math.AG↗

arXiv2013

Normal Functions and the Geometry of Moduli Spaces of Curves

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.

math.AG↗

arXiv2013

Monodromy of Codimension-One Sub-Families of Universal Curves

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.

math.AG↗

arXiv2012

The abelianization of the Johnson kernel

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.

math.GR↗