Ricci-flat metrics on the cone over $\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}$

TL;DR

This paper develops a framework for constructing Ricci-flat metrics on the total space of the canonical bundle over a specific del Pezzo surface, providing explicit deformations and analyzing related geometric structures.

Contribution

It introduces a method to explicitly deform the orthotoric metric on the canonical bundle over the del Pezzo surface and examines the non-existence of certain conformal Killing-Yano forms.

Findings

Explicit first-order deformation of orthotoric metric constructed.

Deformation of the conformal Killing-Yano form does not exist.

Framework applicable to Ricci-flat metric construction on complex surfaces.

Abstract

We describe a framework for constructing the Ricci-flat metrics on the total space of the canonical bundle over $\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}$ (the del Pezzo surface of rank one). We construct explicitly the first-order deformation of the so-called `orthotoric metric' on this manifold. We also show that the deformation of the corresponding conformal Killing-Yano form does not exist.