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Complete cscK Metrics on the Local Models of the Conifold Transition

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In this paper, we construct complete constant scalar curvature Kähler (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.

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Correspondence to Jixiang Fu.

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Communicated by H. Ooguri

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Fu, J., Yau, ST. & Zhou, W. Complete cscK Metrics on the Local Models of the Conifold Transition. Commun. Math. Phys. 335, 1215–1233 (2015).

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