Resolutions of non-regular Ricci-flat Kahler cones

TL;DR

This paper constructs explicit complete Ricci-flat Kähler metrics asymptotic to cones over non-regular Sasaki-Einstein manifolds, expanding the class of known solutions with applications to complex geometry and string theory.

Contribution

It provides new explicit constructions of Ricci-flat Kähler metrics on orbifold fibrations and line bundles over Fano and Sasaki-Einstein manifolds, including non-regular cases.

Findings

Constructed metrics on orbifold fibrations over weighted projective spaces.

Obtained smooth Ricci-flat metrics on vector bundles over V.

Generated metrics on resolutions of cones over Y^{p,q} manifolds.

Abstract

We present explicit constructions of complete Ricci-flat Kahler metrics that are asymptotic to cones over non-regular Sasaki-Einstein manifolds. The metrics are constructed from a complete Kahler-Einstein manifold (V,g_V) of positive Ricci curvature and admit a Hamiltonian two-form of order two. We obtain Ricci-flat Kahler metrics on the total spaces of (i) holomorphic C^2/Z_p orbifold fibrations over V, (ii) holomorphic orbifold fibrations over weighted projective spaces WCP^1, 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 Kahler 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=CP^1 our results give Ricci-flat Kahler orbifold metrics on various toric partial resolutions of the cone over the Sasaki-Einstein manifolds Y^{p,q}.