  The first part of this paper is a generalization of the Feix-Kaledin theorem
on the existence of a hyperkahler metric on a neighbourhood of the zero section
of the cotangent bundle of a Kahler manifold. We show that the problem of
constructing a hyperkahler structure on a neighbourhood of a complex Lagrangian
submanifold in a holomorphic symplectic manifold reduces to the existence of
certain deformations of holomorphic symplectic structures. The Feix-Kaledin
structure is recovered from the twisted cotangent bundle. We then show that
every holomorphic symplectic groupoid over a compact holomorphic Poisson
surface of Kahler type has a hyperkahler structure on a neighbourhood of its
identity section. More generally, we reduce the existence of a hyperkahler
structure on a symplectic realization of a holomorphic Poisson manifold of any
dimension to the existence of certain deformations of holomorphic Poisson
structures adapted from Hitchin's unobstructedness theorem.
