Symplectic resolutions of character varieties

Abstract

In this article, we consider the $G$-character variety of a compact Riemann surface of genus $g > 0$, when $G$ is $\mathrm{SL}(n,\mathbb{C})$ or $\mathrm{GL}(n,\mathbb{C})$. We show that these varieties are symplectic singularities and classify when they admit symplectic resolutions: they do when $g = 1$ or $n = 1$ or $(g,n)=(2,2)$.