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)(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/ZpC2/Zp orbifold fibrations over VV, (ii) holomorphic orbifold fibrations over weighted projective spaces WCP1WCP1, with generic fibres being the canonical complex cone over VV, 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 VV, and (b) the canonical line bundle over a family of geometrically ruled Fano manifolds with base VV. When V=CP1V=CP1 our results give Ricci-flat Kähler orbifold metrics on various toric partial resolutions of the cone over the Sasaki–Einstein manifolds Yp,qYp,q.