Symplectic structure on the character varieties of Sasakian threefolds

Abstract

Take a compact Sasakian threefold $M$ and consider the associated irreducible $\text{SL}(r,\mathbb{C})$-character variety $\mathcal{R}:=\text{Hom}({\pi }_1(M,x_0),\text{SL}(r,\mathbb{C}){)}^{ir}/\text{SL}(r,\mathbb{C})$ of $M$, where $\text{Hom}({\pi }_1(M,x_0),\text{SL}(r,\mathbb{C}){)}^{ir}$ is the space of irreducible homomorphisms. We first construct a natural algebraic $2$-form on $\mathcal{R}$. Then it is shown that this $2$--form is closed. Finally we show that the restriction of this $2$--form to $\text{Hom}({\pi }_1(M,x_0),\text{SU}(r){)}^{ir}$ is symplectic.

Final version