Symplectic structure on the character varieties of Sasakian threefolds

TL;DR

This paper constructs and proves the symplectic nature of a natural algebraic 2-form on the character variety of irreducible representations of the fundamental group of a compact Sasakian threefold.

Contribution

It introduces a natural algebraic 2-form on the character variety and demonstrates its closedness and symplectic restriction on special subvarieties.

Findings

Constructed a natural algebraic 2-form on the character variety.

Proved the 2-form is closed.

Showed the 2-form restricts to a symplectic form on the SU(r) character variety.

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.