Ricci-flat K Ahler Metrics on Canonical Bundles

We prove the existence of a (unique) S-invariant Ricci-flat Kähler metric on a neighbourhood of the zero section in the canonical bundle of a realanalytic Kähler manifold X, extending the metric on X. In the important paper [3], Calabi proved existence of Ricci-flat Kähler metrics on two classes of manifolds: a) cotangent bundles of projective spaces; b) canonical bundles of Kähler-Einstein manifolds. The metrics on T ∗CPn are actually hyperkähler and in the intervening years hyperkähler metrics were shown to exist on cotangent bundles of many other Kähler manifolds. Finally, recently, B. Feix [4] and, independently, D. Kaledin [9] have shown that a real-analytic Kähler metric on a complex manifold X always extends to a (essentially unique) hyperkähler metric on a neighbourhood of X in T ∗X . The aim of this paper is to prove the analogous generalization for the other class of Calabi’s metrics. Our main existence result can be stated as follows: Theorem 1. Let X be a real-analytic Kähler manifold. Then there exists a unique Ricci-flat Kähler metric on a neighbourhood of X in the canonical bundle KX of X which extends the metric on X and for which the standard S-action on KX is isometric and Hamiltonian. The condition of real-analycity of the Kähler metric is clearly necessary, since the extended metric is Ricci-flat. We also notice that the adjunction formula shows that the canonical bundle is the only line bundle over X which can admit a Ricci-flat Kähler metric. 1. Proof of Theorem 1 Let M be an n+ 1 dimensional Kähler manifold with a free Hamiltonian circle action. Then the metric can be locally written in the form: G = ∑ gijdzi ⊗ dz̄j + wdt + w−1φ2, (1.1) where t is the moment map on M , φ is the circle-invariant 1-form and the zi are local coordinates on M/C∗. The complex structure I maps dt to w−1φ. Pedersen and Poon [11] (and LeBrun [10] for n = 1) have worked out the conditions for the complex structure to be integrable and for the metric to be Einstein (in fact, Pedersen and Poon deal with the more general case of torus symmetry). We recall their theorem. Theorem 1.1. [Pedersen-Poon] Let w be a smooth positive funtion and [gij ] a positive definite hermitian matrix of smooth functions on an open set U in Cn×R. Research supported by an EPSRC Advanced Research Fellowship. 1