\(E\)-polynomial of \(SL_{2}(\mathbb{C})\)-character varieties of free groups
type: Item
instance of: scholarly article
zbMATH Open document ID: 1325.14065
DOI: 10.1142/S0129167X1450058X
author: Samuel Cavazos
author: Sean Lawton
published in: International Journal of Mathematics
publication date: 2014-08-08T00:00:00Z
full work available at URL: https://arxiv.org/abs/1401.0228
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 14D20
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14G15
zbMATH DE Number: 6327962
zbMATH Keywords: free group
zbMATH Keywords: conjugacy class
zbMATH Keywords: character variety
zbMATH Keywords: finite field
zbMATH Keywords: \(E\)-polynomial
MaRDI profile type: Publication
arXiv ID: 1401.0228
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cites work: Théorie de Hodge. III
cites work: The topology of moduli spaces of free group representations
cites work: Singularities of free group character varieties
cites work: Mixed Hodge polynomials of character varieties. With an appendix by Nicholas M. Katz.
cites work: Hodge polynomials of \(\mathrm{SL}(2,\mathbb{C})\)-character varieties for curves of small genus
cites work: Geometric reductivity over arbitrary base
cites work: \(SL_n\)-character varieties as spaces of graphs
cites work: Character varieties
title: \(E\)-polynomial of \(SL_{2}(\mathbb{C})\)-character varieties of free groups
review text: The Betti cohomologies \(H^\bullet_c(X)\) of compact support of a complex affine variety \(X\) come equipped with a mixed Hodge structure, therefore one can define the mixed Hodge numbers \(h^{p,q;j}_c(X) := \dim_{\mathbb{C}} \mathrm{Gr}^F_p \mathrm{Gr}_{p+q}^{W \otimes \mathbb{C}} H^j_c(X)\), and consequently the mixed Hodge polynomial \(H_c(X; x,y,t) := \sum_{p,q,j} h^{p,q;j}_c(X) x^p y^q t^j\). The \(E\)-polynomial of \(X\) is then defined as \(E(X; x,y) := H_c(X; x,y,-1)\). By spreading-out \(X\) to a scheme \(\mathcal{X}\) over a \(\mathbb{Z}\)-algebra \(R\) in a reasonable way, one says that \(X\) has polynomial count if there exists a polynomial \(P_X \in \mathbb{Z}[X]\) such that for each \(\phi: R \to \mathbb{F}_q\), where \(q\) is a power of \(p\) for all but finitely many primes \(p\), we have \(|\mathcal{X}(\mathbb{F}_q)| = P_X(q)\). It is shown by Katz (appendix to [\textit{T. Hausel} and \textit{F. Rodriguez-Villegas}, Invent. Math. 174, No. 3, 555--624 (2008; Zbl 1213.14020)]) that having polynomial count implies \(E(X; x,y) = P_X(xy)\).NEWLINENEWLINEIn this paper, the authors consider the character varieties \(\mathfrak{X}_\Gamma(G) = \mathrm{Hom}(\Gamma, G) /\!/ G\) (the GIT quotient, \(G\) acts by conjugation on \(\mathrm{Hom}(\Gamma, G)\)) in the case where \(G = \mathrm{SL}_2(\mathbb{C})\) and \(\Gamma\) is the free group of rank \(r\). Such varieties play a prominent role in geometry. Given Katz's result alluded to above, the main idea is to study its \(E\)-polynomial by counting points over various \(\mathbb{F}_q\). Since the conjugation action of \(G\) on \(\mathrm{Hom}(\Gamma, G)\) is not free, one proceeds by stratifying \(\mathrm{Hom}(\Gamma, G)\), a careful yet elementary analysis is required. An explicit form of \(E\) is given in the Theorem B.
reviewed by: Wen-Wei Li
description: scientific article; zbMATH DE number 6327962