Mixed Hodge polynomials of character varieties

By Tamás Hausel, Fernando Rodriguez-Villegas. 2006

310 citations recorded by the source corpus.

We calculate the E-polynomials of certain twisted GL(n,ℂ)-character varieties $\mathcal{M}_{n}$ of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lie-type $\text{GL}(n,\mathbb{F}_{q})$ and a theorem proved in the appendix by N. Katz. We deduce from this calculation several geometric results, for example, the value of the topological Euler characteristic of the associated PGL(n,ℂ)-character variety. The calculation also leads to several conjectures about the cohomology of $\mathcal{M}_{n}$: an explicit conjecture for its mixed Hodge polynomial; a conjectured curious hard Lefschetz theorem and a conjecture relating the pure part to absolutely indecomposable representations of a certain quiver. We prove these conjectures for n=2.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/380e4d0703fdff1197b4d8e5caf1066ffaf5a1a8

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Indecomposable vector bundles and stable Higgs bundles over smooth projective curves (2014, methodology): “al counting geometrically indecomposable representation of Σg over Fq of dimension r(see [K1]). It is conjectured that Ag,r,d(0) = AΣg,r(1)– this for instance would follow from the main conjecture in [HRV1]. However, just as our formula (1.1) slightly diﬀers from that conjectured in [HRV1], so does ”
Poincaré polynomials of character varieties, Macdonald polynomials and affine Springer fibers (2017, unknown): “ points). The case of rank 3 with marked points was studied by Garc´ıa-Prada, Gothen and Mun˜oz in [GPGMn07]. More partial results were obtained by Garc´ıa-Prada, Heinloth and Schmitt in [GPHS11]. In [HRV08] Hausel and Rodriguez-Villegas computed the E-polynomials of character varieties without mark”
Global topology of the Hitchin system (2011, methodology): “One can prove [47] that if d and n are coprime, MB and M̌ d B are non-singular, and so M̂B is an orbifold.”
Topology of Hitchin systems and Hodge theory of character varieties (2010, methodology): “If D = KC , thanks to the results of [32] described in Section 1.”
The integrality conjecture and the cohomology of preprojective stacks (2016, unknown): “r nonzero d ∈ Λζ θ, where Λζ θ ⊂ N Q 0 is as in (15) the submonoid of dimension vectors of slope θ, deﬁne the monodromic mixed Hodge module BPSζ Q,W,d ∈ MMHM M(Q)ζ -ss d ⊂ MMHM M(Q)ζ -ss θ by (17) BPSζ Q,W,d = ( φmon T r(W)ζ d ICM(Q)ζ -ss d (Q) if M(Q)ζ -st d 6= ∅ 0 otherwise, and deﬁne BPSζ Q,W”
Hitchin fibrations, abelian surfaces, and the P=W conjecture (2019, background): “A striking prediction, suggested by the parallel between the Relative Hard Lefschetz [7] and Curious Hard Lefschetz [29] Theorems, was suggested by de Cataldo, Hausel, and Migliorini [13]; it predicts that the perverse filtration of MDol with respect to the Hitchin fibration (2) matches the weight f”
Geometry of Multiplicative Preprojective Algebra (2007, background): “In this direction, he and RodriguezVillegas [15] suggested several interesting conjectures for the mixed Hodge polynomial of twisted character varieties of compact Riemann surfaces.”
On the motives of moduli of chains and Higgs bundles (2011, methodology): “We implemented this formula in Maple and checked that for genus ≤ 21 the result confirms the conjecture of Hausel and Rodŕıguez-Villegas ([17]) on the Poincaré polynomial of this space.”
Hodge polynomials of SL (2,C)-character varieties for curves of small genus (2013, background): “In [14] they obtained the Epolynomials of Md(G) for G = GL(n,C) in terms of a simple generating function.”
Arithmetic harmonic analysis on character and quiver varieties II (2008, background): “For details on the hook function we refer the reader to [12].”