Resolutions of non-regular Ricci-flat Kähler cones

We present explicit constructions of complete Ricci-flat Kähler metrics that are asymptotic to cones over non-regular Sasaki-Einstein manifolds. The metrics are constructed from a complete Kähler-Einstein manifold (V, gV) of positive Ricci curvature and admit a Hamiltonian two-form of order two. We obtain Ricci-flat Kähler metrics on the total spaces of (i) holomorphic C2 / Zp orbifold fibrations over V, (ii) holomorphic orbifold fibrations over weighted projective spaces W C P1, with generic fibres being the canonical complex cone over V, and (iii) the canonical orbifold line bundle over a family of Fano orbifolds. As special cases, we also obtain smooth complete Ricci-flat Kähler metrics on the total spaces of (a) rank two holomorphic vector bundles over V, and (b) the canonical line bundle over a family of geometrically ruled Fano manifolds with base V. When V = C P1 our results give Ricci-flat Kähler orbifold metrics on various toric partial resolutions of the cone over the Sasaki-Einstein manifolds Yp, q. © 2009 Elsevier B.V. All rights reserved.