Complete cscK Metrics on the Local Models of the Conifold Transition

Abstract

In this paper, we construct complete constant scalar curvature Kahler (cscK) metrics on the complement of the zero section in the total space of \({\mathcal{O}(-1)^{\oplus2}}\) over \({\mathbb{P}^{1}}\), which is biholomorphic to the smooth part of the cone C 0 in \({\mathbb{C}^{4}}\) defined by equation \({\Sigma_{i=1}^{4} w_{i}^{2}=0}\). On its small resolution and its deformation, we also consider complete cscK metrics and find that if the cscK metrics are homogeneous, then they must be Ricci-flat.