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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.

Using twistor techniques we shall show that there is a hypercomplex structure in the neighbourhood of the zero section of the tangent bundle TX of any complex manifold X with a real-analytic torsion-free connection compatible with the complex structure whose curvature is of type (1, 1). The zero section is totally geodesic and the Obata connection restricts to the given connection on the zero section.We also prove an analogous result for vector bundles: any vector bundle with real-analytic connection whose curvature is of type (1, 1) over X can be extended to a hyperholomorphic bundle over a neighbourhood of the zero section of TX.