Mixed Hodge polynomials of character varieties

Hausel, Tamás; Rodríguez-Villegas, Fernando

doi:10.1007/s00222-008-0142-x

Cited by 233 publications

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“…On the other hand, a curious hard Lefschetz theorem is conjectured in [32,Conj. 4.2.7] to hold for the character variety for PGL n , which would of course follow, if P = W , from the relative hard Lefschetz theorem.…”

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confidence: 99%

“…Character variety. In this section, we recall some definitions and results from [32]. Throughout the paper, the singular homology and cohomology groups are taken with rational coefficients.…”

Section: 2mentioning

confidence: 99%

“…The paper [32] investigates in depth one of the important algebro-geometric invariants of M B , namely the mixed Hodge structure on its cohomology groups. In view of [32,Cor.…”

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Topology of Hitchin systems and Hodge theory of character varieties: the case A_1

2012

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For G = GL2, PGL2, SL2 we prove that the perverse filtration associated with the Hitchin map on the rational cohomology of the moduli space of twisted G-Higgs bundles on a compact Riemann surface C agrees with the weight filtration on the rational cohomology of the twisted G character variety of C when the cohomologies are identified via non-Abelian Hodge theory. The proof is accomplished by means of a study of the topology of the Hitchin map over the locus of integral spectral curves.

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Topology of Hitchin systems and Hodge theory of character varieties: the case A_1

2012

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“…For more general applications (to quiver and character varieties) we refer the reader to [1] and [2].…”

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“…a function called degree such that (1) there are finitely many colors of a given degree; (2) there is a unique color 0 ∈ C of degree 0. A coloring on X is a map…”

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Counting colorings on varieties

Rodríguez-Villegas¹

2007

Publicacions Matemàtiques

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1.1. Introduction. The goal of this note is to present a combinatorial mechanism for counting certain objects associated to a variety X defined over a finite field. The basic example, discussed in §2.2, is that of counting conjugacy classes in GL n (F q ), where X = G m (the multiplicative group).We give four different forms of the main formula (which is somewhat reminiscent of Polya's theory of counting). The principle that emerges is that in a given setup the counting generating functions for X = • (a point), X = G a (the additive group) and X = G m are related to one another in a simple way. Often one of the cases will be significantly easier to compute than the others yielding a closed formula for all three generating functions. For example, in §3 we describe how one can go from counting all matrices in M n (F q ), corresponding to X = G a , to counting unipotent matrices in M n (F q ), corresponding to X = •.None of the special cases considered here are really new; the point is, instead, to stress the main combinatorial principle. For more general applications (to quiver and character varieties) we refer the reader to [1] and [2].1.2. The Zeta Function of X. Let F q be a finite field with q elements. Fix an algebraic closure F q of F q . For each r ∈ N let F q r be the unique subfield of F q of cardinality q r . Let Frob q ∈ Gal(F q /F q ) be the Frobenius automorphism x → x q . Then F q r is the fixed field of Frob r q . Let X be an algebraic variety defined over F q . For each r ∈ N let N r (X) := #X(F q r ). The zeta function of X is defined as

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Projective spaces over

Thas

2018

J of Combinatorial Designs

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In this essay, we study various notions of projective space (and other schemes) over F 1 ℓ, with F 1 denoting the field with one element. Our leading motivation is the “Hidden Points Principle,” which shows a huge deviation between the set of rational points as closed points defined over F 1 ℓ and the set of rational points defined as morphisms monospace𝚂𝚙𝚎𝚌 ( double-struckF 1 ℓ ) ↦ scriptX. We also introduce, in the same vein as Kurokawa [Proc. Jpn. Acad. Ser. A Math. Sci. 81 (2005), pp. 180–184], schemes of F 1 ℓ‐type and consider their zeta functions.

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Mixed Hodge polynomials of character varieties

Cited by 233 publications

References 57 publications

Topology of Hitchin systems and Hodge theory of character varieties: the case A_1

Topology of Hitchin systems and Hodge theory of character varieties: the case A_1

Counting colorings on varieties

Projective spaces over

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