E-polynomials of the \(\mathrm{SL}(2, \mathbb{C})\)-character varieties of surface groups (Q2796768)

The main goal of this work is to explicitly determine the \(E\)-polynomials of the \(\mathrm{SL}(2,\mathbb{C})\)-character variety of a smooth complex projective curve \(X\), of positive genus \(g\), with one puncture, and with any given fixed holonomy around the puncture.NEWLINENEWLINETo define these spaces, fix an arbitrary point \(p_0\in X\). The \(\mathrm{SL}(2,\mathbb{C})\)-character variety is the moduli space \(\mathcal{M}_C(\mathrm{SL}(2,\mathbb{C}))\) of semisimple representations \(\rho:\pi_1(X\setminus\{p_0\})\to\mathrm{SL}(2,\mathbb{C})\) such that \(\rho(\gamma)=C\), where \(\gamma\) is a small loop around \(p_0\) and \(C\) is any element in \(\mathrm{SL}(2,\mathbb{C})\). This can be defined as the GIT quotient NEWLINE\[NEWLINE\mathcal{M}_C:=\mathcal{M}_C(\mathrm{SL}(2,\mathbb{C}))=\left\{(A_1,B_1,\ldots,A_g,B_g)\in\mathrm{SL}(2,\mathbb{C})^{2g}\,|\,\prod_{i=1}^g[A_i,B_i]=C\right\}//\;\mathrm{Stab}(C).NEWLINE\]NEWLINE Notice that this space only depends on the conjugacy class of the holonomy \(C\). Notice also that \(C\) may not be diagonalisable. When it is, the space \(\mathcal{M}_C\) is homeomorphic to the moduli space of (parabolic) \(\mathrm{SL}(2,\mathbb{C})\)-Higgs bundles over \(X\), although this homeomorphism is not algebraic.NEWLINENEWLINEWrite \(e(\mathcal{M}_C)\) for the \(E\)-polynomial of \(\mathcal{M}_C\). Its computation uses the geometric technique introduced by \textit{M. Logares} et al. [Rev. Mat. Complut. 26, No. 2, 635--703 (2013; Zbl 1334.14006)] by using suitable stratifications of \(\mathcal{M}_C\) as well as handling fibrations which are locally trivial in the analytic topology but not in the Zariski topology. The main results of loc. cit. yielded explicit formulas for the \(E\)-polynomials when \(g=1,2\). In [Osaka J. Math. 53, No. 3, 645--681 (2016; Zbl 1369.14014)], the authors extended the theory to the case \(g=3\), and holonomy \(C=\pm\mathrm{Id}\). Thus this paper completes the general case \(g\geq 3\) and any holonomy.NEWLINENEWLINEThe first result is that, for any \(C\), the mixed Hodge structure \(H^{k;p,q}_c(\mathcal{M}_C)\) is non-zero only when \(p=q\), that is \(\mathcal{M}_C\) is of balanced type.NEWLINENEWLINEThen the main theorem is the explicit computation of \(e(\mathcal{M}_{\mathrm{Id}})\), \(e(\mathcal{M}_{\mathrm{-Id}})\), \(e(\mathcal{M}_{J_+})\), with \(J_+=\left(\begin{smallmatrix} 1 & 1 \\0 & 1\end{smallmatrix}\right)\), \(e(\mathcal{M}_{J_-})\), with \(J_-=\left(\begin{smallmatrix} -1 & 1 \\0 & -1\end{smallmatrix}\right)\) and \(e(\mathcal{M}_{\xi_\lambda})\), with \(\xi_\lambda=\left(\begin{smallmatrix}\lambda & 0 \\0 & \lambda^{-1}\end{smallmatrix}\right)\) and \(\lambda\in\mathbb{C}\setminus\{0,\pm 1\}\).NEWLINENEWLINEThe polynomial \(e(\mathcal{M}_{\mathrm{-Id}})\) has been also explicitly determined through arithmetic methods in [\textit{M. Mereb}, Math. Ann. 363, No. 3--4, 857--892 (2015; Zbl 1333.14041)].NEWLINENEWLINEIt follows directly from the main theorem that the relation \(e(\mathcal{M}_{\xi_\lambda})=e(\mathcal{M}_{J_-})+(q+1)e(\mathcal{M}_{\mathrm{-Id}})\) holds (where \(q=uv\) and \(u,v\) are the variables of the \(E\)-polynomial), as has been conjectured in [Zbl 1334.14006].NEWLINENEWLINEAnother direct consequence is that the \(E\)-polynomials of \(\mathcal{M}_{-\mathrm{Id}}\) and \(\mathcal{M}_{\xi_\lambda}\) are palindromic. As a final byproduct of their study, the authors also obtain the behavior of \(e(\mathcal{M}_{\xi_\lambda})\) when one varies the parameter \(\lambda\in\mathbb{C}\setminus\{0,\pm 1\}\). It turns out that the monodromy around \(\pm 1\) while the one around \(0\) is of order \(2\). The behavior \(e(\mathcal{M}_{\xi_\lambda})\) with varying \(\lambda\) is encoded in a polynomial \(R(\mathcal{M}_{\xi_\lambda})\in R(\mathbb{Z}_2)[q]\) with coefficients in the representation ring \(R(\mathbb{Z}_2)\). This ring has two generators, namely the trivial representation \(T\) and the non-trivial one \(N\). The explicit description of \(R(\mathcal{M}_{\xi_\lambda})\) in terms of \(T\) and \(N\) is obtained. The coefficient of \(T\) is the invariant part of the cohomology of \(\mathcal{M}_{\xi_\lambda}\), under the natural \(\mathbb{Z}_2\) action, while the coefficient of \(N\) is the ``variant'' part. This computation is important for mirror symmetry, since it relates with the space \(\mathcal{M}_{\xi_\lambda}(\mathrm{PGL}(2,\mathbb{C}))\), recalling that \(\mathrm{SL}(2,\mathbb{C})\) and \(\mathrm{PGL}(2,\mathbb{C})\) are Langlands dual groups.