Mixed Hodge polynomials of character varieties. With an appendix by Nicholas M. Katz.
type: Item
instance of: scholarly article
title: Mixed Hodge polynomials of character varieties. With an appendix by Nicholas M. Katz.
zbMATH Open document ID: 1213.14020
author: Tamás Hausel
author: Fernando Rodriguez-Villegas
published in: Inventiones Mathematicae
publication date: 2009-03-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0612668
review text: Let \(g\geq 0\) and \(n>0\) be integers. Let \(\zeta_n\in \mathbb C\) be a primitive \(n\)-th root of unity. Let \(I_n\) be the identity \(n\times n\)-matrix. For two square matrices \(A\) and \(B\) denote their commutator \(ABA^{-1}B^{-1}\) by \([A, B]\) and define \(\mathcal M_n\) to be the following affine GIT quotient by the conjugation action of \(\mathrm{GL}(n, \mathbb C)\): \[ \{ (A_1, B_1,\dots, A_g, B_g) \mid A_i, B_i\in \mathrm{GL}(n, \mathbb C),\quad [A_1, B_1]\cdot\dots\cdot[A_g, B_g]=\zeta_n I_n \}// \mathrm{GL}(n, \mathbb C). \] \(\mathcal M_n\) is a twisted character variety of a genus \(g\) closed Riemann surface. The paper under review deals with such varieties. The \(E\)-polynomials of \(\mathcal{M}_{n}\) are calculated by counting points over finite fields using the character table of the finite group of Lie-type \(\mathrm{GL}(n,\mathbb{F}_{q})\) and a theorem proved in the appendix of the paper by N. Katz. Several geometric results are deduced from this calculation, for example, the value of the topological Euler characteristic of the associated \(\mathrm{PGL}(n,\mathbb C)\)-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. These conjectures are proven for \(n=2\). The paper consists of 6 sections. 1. A motivation with relevant references is presented in the introduction. An outline of the paper is given, the main results of the paper are formulated. 2. In Section~2 various facts needed later are collected. 3. Section~3 deals with the calculation of the \(E\)-polynomial of \(\mathcal M_n\). 4. In Section~4 a conjecture on the mixed Hodge polynomial of \(\mathcal M_n\) is formulated, some consequences are derived. 5. The proof of all the conjectures from Section~4 is given in Sectio…
reviewed by: Oleksandr Iena
Mathematics Subject Classification ID: 14D07
Mathematics Subject Classification ID: 14C30
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 14H60
Mathematics Subject Classification ID: 14D99
zbMATH DE Number: 5530512
zbMATH Keywords: character varieties
zbMATH Keywords: \(E\)-polynomials
zbMATH Keywords: Hodge structures
zbMATH Keywords: mixed Hodge polynomials
zbMATH Keywords: Poincaré polynomial
zbMATH Keywords: zeta-equivalence
MaRDI profile type: Publication
OpenAlex ID: W2114120297
arXiv ID: math/0612668
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Mixed Hodge polynomials of character varieties. With an appendix by Nicholas M. Katz. (cites work)
cites work: Q4860516

Mixed Hodge polynomials of character varieties. With an appendix by Nicholas M. Katz. (DOI)
DOI: 10.1007/S00222-008-0142-X