Hyperkahler metrics on cotangent bundles

For any real-analytic Kahler manifold we shall prove the existence of a hyperkahler metric in a neighbourhood of the zero section of its cotangent bundle using the twister method. This hyperkahler metric is compatible with the canonical holomorphic-symplectic structure of the cotangent bundle, extends the given Kahler metric, and the circle action given by scalar multiplication in the fibres is isometric. By deriving necessary conditions we show that many of these hyperkahler metrics will be incomplete.