  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 $M \subset T^*M$ 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 $M \subset T^*M$. 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.

Algebraic Geometry;Algebraic Geometry