Following A. Beauville, a complex algebraic variety $X$ is said to be symplectic if it admits a holomorphic symplectic form $\omega$ on its smooth locus such that, for every resolution $\pi: Y \to X$, $\pi^*\omega$ extends to a holomorphic $2$-form on $Y$. When this extension is actually non-degenerate (a de facto symplectic form) on $Y$, we call $\pi$ a symplectic (or crepant) resolution.

Let $G$ be a complex reductive group and $A$ an abelian variety of dimension $d$. The aim of this talk is to show that all moduli spaces of $G$-Higgs bundles over $A$ are symplectic varieties, and that, for $G=\mathrm{GL}(n,\mathbb C)$, the canonical Hilbert-Chow morphism is a symplectic resolution if and only if $d=1$.

Moreover, using a little representation theory, we can obtain explicit expressions for the Poincaré polynomials of all Hilbert-Chow resolutions (either $d=1$, all $n$; or $n=1,2,3$ and all $d$). This is joint work with I. Biswas and A. Nozad.