A natural closed algebraic 2-form is defined on the SL(r,C) character variety of Sasakian threefolds that restricts to a symplectic form on the SU(r) locus.
Kasuya, Non-abelian Hodge correspondence and moduli spaces of flat bundles on Sasakian manifolds with fixed basic structures, preprint
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Symplectic structure on the character varieties of Sasakian threefolds
A natural closed algebraic 2-form is defined on the SL(r,C) character variety of Sasakian threefolds that restricts to a symplectic form on the SU(r) locus.