Journal ArticleJournal of the European Mathematical Society · January 1, 2021
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 a quasi-algebraic framing is a morphism of mixed Hodge struc ...
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Journal ArticleEMS Surveys in Mathematical Sciences · January 1, 2021
Torelli groups are subgroups of mapping class groups that consist of those diffeomorphism classes that act trivially on the homology of the associated closed surface. The Johnson homomorphism, defined by Dennis Johnson, and its generalization, defined by S ...
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Journal ArticleJournal of the Institute of Mathematics of Jussieu · May 1, 2020
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In this paper we construct a-linear tannakian category of universal mixed elliptic motives over the moduli space of elliptic curves. It contains , the category of mixed Tate motives unramified over the integers. Each object of is an object of endowed with ...
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Journal ArticlePure and Applied Mathematics Quarterly · January 30, 2020
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 writ ...
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Journal ArticlePure and Applied Mathematics Quarterly · January 1, 2020
In this paper, we give an exposition of the elliptic KZB connection over the universal elliptic curve and use it to compute the limit mixed Hodge structure on the unipotent fundamental group of the first order Tate curve. We also give an explicit algebraic ...
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Journal ArticleAlgebra and Number Theory · January 1, 2018
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 ...
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Chapter · 2017
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 comp ...
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Journal ArticleCommunications in Contemporary Mathematics · August 1, 2016
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 ...
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Journal ArticleJournal of the European Mathematical Society · January 1, 2014
We prove that the first complex homology of the Johnson subgroup of the Torelli group Tg is a non-trivial, unipotent Tg-module for all g ≥ 4 and give an explicit presentation of it as a Sym H 1(Tg,C)-module when g ≥ 6. We do this by proving that, for a fin ...
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Journal ArticleJournal of Algebraic Geometry · June 18, 2013
In this paper we develop a theory of non-abelian cohomology for proalgebraic groups which is used in J. Amer. Math. Soc. 24 (2011), 709-769 to study the unipotent section conjecture. The non-abelian cohomology H 1nab(G,P) is a scheme. The argument G is a p ...
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Journal ArticleJournal of the European Mathematical Society · January 1, 2021
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 a quasi-algebraic framing is a morphism of mixed Hodge struc ...
Full textOpen AccessCite
Journal ArticleEMS Surveys in Mathematical Sciences · January 1, 2021
Torelli groups are subgroups of mapping class groups that consist of those diffeomorphism classes that act trivially on the homology of the associated closed surface. The Johnson homomorphism, defined by Dennis Johnson, and its generalization, defined by S ...
Full textCite
Journal ArticleJournal of the Institute of Mathematics of Jussieu · May 1, 2020
Featured Publication
In this paper we construct a-linear tannakian category of universal mixed elliptic motives over the moduli space of elliptic curves. It contains , the category of mixed Tate motives unramified over the integers. Each object of is an object of endowed with ...
Full textCite
Journal ArticlePure and Applied Mathematics Quarterly · January 30, 2020
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 writ ...
Open AccessLink to itemCite
Journal ArticlePure and Applied Mathematics Quarterly · January 1, 2020
In this paper, we give an exposition of the elliptic KZB connection over the universal elliptic curve and use it to compute the limit mixed Hodge structure on the unipotent fundamental group of the first order Tate curve. We also give an explicit algebraic ...
Full textCite
Journal ArticleAlgebra and Number Theory · January 1, 2018
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 ...
Full textCite
Chapter · 2017
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 comp ...
Link to itemCite
Journal ArticleCommunications in Contemporary Mathematics · August 1, 2016
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 ...
Full textCite
Journal ArticleJournal of the European Mathematical Society · January 1, 2014
We prove that the first complex homology of the Johnson subgroup of the Torelli group Tg is a non-trivial, unipotent Tg-module for all g ≥ 4 and give an explicit presentation of it as a Sym H 1(Tg,C)-module when g ≥ 6. We do this by proving that, for a fin ...
Full textOpen AccessCite
Journal ArticleJournal of Algebraic Geometry · June 18, 2013
In this paper we develop a theory of non-abelian cohomology for proalgebraic groups which is used in J. Amer. Math. Soc. 24 (2011), 709-769 to study the unipotent section conjecture. The non-abelian cohomology H 1nab(G,P) is a scheme. The argument G is a p ...
Full textCite
Journal ArticleJournal of the Institute of Mathematics of Jussieu · January 1, 2005
Suppose that [formula omitted] is a subfield of [formula omitted] for which the [formula omitted] -adic cyclotomic character has infinite image. Suppose that [formula omitted] is a curve of genus [formula omitted] defined over [formula omitted], and that [ ...
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Journal ArticleMathematical Research Letters · January 1, 2005
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 giv ...
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Journal ArticleJournal of Differential Geometry · January 1, 2004
In this paper we compute the asymptotics of the natural metric on the line bundle over the moduli spaceMg associated to the algebraic cycle C − C− in the jacobian Jac C of a smooth projective curve C of genus g ≤ 3. The asymptotics are related to the struc ...
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Journal ArticleCompositio Mathematica · November 1, 2003
Fix a prime number l. We prove a conjecture stated by Ihara, which he attributes to Deligne, about the action of the absolute Galois group on the pro-l completion of the fundamental group of the thrice punctured projective line. Similar techniques are also ...
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Journal ArticleAnnales Scientifiques de l'Ecole Normale Superieure · 1998
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Suppose that X is a smooth manifold and ρ : π1 (X,N) → S is a representation of the fundamental group of X into a real reductive group with Zariski dense image. To such data one can associate the Malcev completion G of π1(X,x) relative to ρ. In this paper ...
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Journal Article$K$-Theory. An Interdisciplinary Journal for the Development, Application, and Influence of $K$-Theory in the Mathematical Sciences · 1987Full textCite
Journal ArticleK-Theory · 1987
In this paper we use Chen's iterated integrals to put a mixed Hodge structure on the homotopy Lie algebra of an arbitrary complex algebraic variety, generalizing work of Deligne and Morgan. Similar techniques are used to put a mixed Hodge structure on othe ...
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Journal ArticleProceedings of the American Mathematical Society · January 1, 1986
An explicit formula for a canonical splitting s: Qℬ(ℰ.)⟶ℬ(ℰ.) of the projection ℬ(ℰ.)⟶Qℬ(ℰ.) of the bar construction on a commutative d.g. algebraℰ.onto its indecomposables is given. We prove that s induces a d.g. algebra isomorphism Λ(Qℬ(ℰ.))⟶ℬ(ℰ.) and th ...
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Journal ArticleProceedings of the American Mathematical Society · 1979
A sufficient condition for a function defined on a Banach space to be
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