  In this article we consider the connected component of the identity of
$G$-character varieties of compact Riemann surfaces of genus $g > 0$, for
connected complex reductive groups $G$ of type $A$ (e.g., $SL_n$ and $GL_n$).
We show that these varieties are symplectic singularities and classify which
admit symplectic resolutions. The classification reduces to the semi-simple
case, where we show that a resolution exists if and only if either $g=1$ and
$G$ is a product of special linear groups of any rank and copies of the group
$PGL_2$, or if $g=2$ and $G = (SL_2)^m$ for some $m$.
