Abstract

I will discuss the existence of hyperkähler structures on symplectic realizations of holomorphic Poisson manifolds, and show that they always exist when the Poisson manifold has complex dimension two. We obtain this structure by constructing a twistor space by lifting special deformations of the Poisson surface adapted from Hitchin's unobstructedness theorem. In the case of the zero Poisson structure, we recover the Feix-Kaledin hyperkähler structure on the cotangent bundle of a Kähler manifold. This talk is based on arXiv:2011.09282.