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.