On Character Variety of Anosov Representations

Let $\mathrm{\Gamma }$ be the fundamental group of a $k$-punctured, $k\ge 0$, closed connected orientable surface of genus $g\ge 2$. We show that the character variety of the $(Q^{+},Q^{-})$-Anosov irreducible representations, resp. the character variety of the $(P^{+},P^{-})$-Anosov Zariski dense representations of $\mathrm{\Gamma }$ into $\text{SL}(n,\text{C})$, $n\ge 2$, is a complex manifold of complex dimension \hbox{$(2g+k-2)(n^2-1)$}. For $\mathrm{\Gamma }={\pi }_1({\mathrm{\Sigma }}_g)$, we also show that these character varieties are holomorphic symplectic manifolds.