Hyperkaehler structures on total spaces of holomorphic cotangent bundles

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Let M be a Kaehler manifold, and consider the total space T*M of the cotangent bundle to M. We show that in the formal neighborhood of the zero section M ⊂ T*M the space T*M admits a canonical hyperkaehler structure, compatible with the complex and holomorphic symplectic structures on T*M. The associated hyperkaehler metric h coincides with the given Kaehler metric on the zero section M ⊂ T*M. Moreover, h is invariant under the canonical circle action on T*M by dilatations along the fibers of T*M over M. We show that a hyperkaehler structure with these properties is unique. When the Kaehler metric on M is real-analytic, we show that this formal hyperkaehler structure can be extended to an open neighborhood of the zero section. We also prove a hyperkaehler analog of the Darboux-Weinstein Theorem. To prove these results, we use the machinery of R-Hodge structures, following Deligne and Simpson.