Twistor spaces of hyperkähler manifolds with S1-actions

We shall describe the twistor space of a hyperkahler 4 n -manifold with an isometric S 1 -action which is holomorphic for one of the complex structures, scales the corresponding holomorphic symplectic form and whose fixed point set has complex dimension n . We deduce that any hyperkahler metric on the cotangent bundle of a real-analytic Kahler manifold which is compatible with the canonical holomorphic symplectic structure, extends the given Kahler metric and for which the S 1 -action by scalar multiplication in the fibres is isometric is unique in a neighbourhood of the zero section. These metrics have been constructed independently by the author and Kaledin.