Mirror symmetry and Langlands duality in the non-Abelian Hodge theory of a curve
TLDR
The mirror symmetry conjecture of Hausel-Rodriguez-Villegas as discussed by the authors is related to the mirror symmetry conjectures of Hitchin, Gothen, Garsia-Haiman and Earl-Kirwan.Abstract:
The paper surveys the mirror symmetry conjectures of Hausel-Thaddeus and Hausel-Rodriguez-Villegas concerning the equality of certain Hodge numbers of SL(n, ℂ) vs. PGL(n, ℂ) flat connections and character varieties for curves, respectively. Several new results and conjectures and their relations to works of Hitchin, Gothen, Garsia-Haiman and Earl-Kirwan are explained. These use the representation theory of finite groups of Lie-type via the arithmetic of character varieties and lead to an unexpected conjecture for a Hard Lefschetz theorem for their cohomology.read more
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Journal ArticleDOI
Mixed Hodge polynomials of character varieties
TL;DR: In this article, the E-polynomials of certain twisted GL(n,ℂ)-character varieties of Riemann surfaces were calculated by counting points over finite fields using the character table of the finite group of Lie-type.
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Mixed Hodge polynomials of character varieties
TL;DR: In this paper, the E-polynomials of certain twisted GL(n,C)-character varieties M_n of Riemann surfaces were calculated by counting points over finite fields using the character table of the finite group of Lie-type character varieties.
Book
Moduli of representations of the fundamental group of a smooth projective variety, II . Finite group actions and étale cohomology . A rank theorem for analytic maps between power series spaces . Some groups whose reduced C[*]-algebra is simple . Existence de 1-formes fermées non singulières dans une classe de cohomologie de de Rahm
Carlos Simpson,Jeremy Rickard,Herwig Hauser,Gerd Müller,M. Bekka,Michael Cowling,Pierre de la Harpe,François Latour +7 more
TL;DR: In this paper, the authors present a legal opinion on the applicability of commercial or impression systématiques in the context of the agreement of publication mathématique de l'I.H.S.
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Topology of Hitchin systems and Hodge theory of character varieties: the case A1
TL;DR: In this article, the Hitchin map on the moduli space of twisted G-Higgs bundles on a compact Riemann surface was shown to agree with the weight ltration on the rational cohomology of the twisted G character variety of C when the cohomologies are identied via non-Abelian Hodge theory.
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Topology of Hitchin systems and Hodge theory of character varieties
TL;DR: In this article, the Hitchin map on the moduli space of twisted G-Higgs bundles on a Riemann surface has been shown to agree with the weight filtration on the cohomology of the twisted G character variety.
References
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Principles of Algebraic Geometry
Phillip Griffiths,Joe Harris +1 more
TL;DR: In this paper, a comprehensive, self-contained treatment of complex manifold theory is presented, focusing on results applicable to projective varieties, and including discussion of the theory of Riemann surfaces and algebraic curves, algebraic surfaces and the quadric line complex.
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The Yang-Mills equations over Riemann surfaces
Michael Atiyah,Raoul Bott +1 more
TL;DR: In this article, the Yang-Mills functional over a Riemann surface is studied from the point of view of Morse theory, and the main result is that this is a perfect 9 functional provided due account is taken of its gauge symmetry.
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The self-duality equations on a riemann surface
TL;DR: In this paper, the authors studied a special class of solutions of the self-dual Yang-Mills equations on Riemann surfaces and showed that the moduli space of all solutions turns out to be a manifold with an extremely rich geometric structure.
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Théorie de Hodge : III
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Mirror symmetry is T duality
TL;DR: In this paper, it was argued that every Calabi-Yau manifold X with a mirror Y admits a family of supersymmetric toroidal 3-cycles and that the moduli space of such cycles together with their flat connections is precisely the space Y.