\(E\)-polynomial of \(SL_{2}(\mathbb{C})\)-character varieties of free groups
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.
scientific article; zbMATH DE number 6327962
scientific article