Quasi-finiteness of morphisms between character varieties

Abstract

Let $f:Y\to X$ be a morphism between smooth complex quasi-projective varieties and $Z$ be the closure of $f(Y)$ with $\iota :Z\to X$ the inclusion map. We prove that a. for any field $K$, there exist finitely many semisimple representations $\{{\tau }_i:{\pi }_1(Z)\to {\mathrm{GL}}_N(\overline{k}){\}}_{i=1,\dots ,\ell }$ with $k\subset K$ the minimal field contained in $K$ such that if $\varrho :{\pi }_1(X)\to {\mathrm{GL}}_N(K)$ is any representation satisfying $[f^{\ast }\varrho ]=1$, then $[{\iota }^{\ast }\varrho ]=[{\tau }_i]$ for some $i$. b. The induced morphism between ${\mathrm{GL}}_N$-character varieties (of any characteristic) of ${\pi }_1(X)$ and ${\pi }_1(Y)$ is quasi-finite if $\mathrm{Im}[{\pi }_1(Z)\to {\pi }_1(X)]$ is a finite index subgroup of ${\pi }_1(X)$. These results extend the main results by Lasell in 1995 and Lasell-Ramachandran in 1996 from smooth complex projective varieties to quasi-projective cases with richer structures.

10 pages