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Complete cscK Metrics on the Local Models of the Conifold Transition

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Abstract

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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Authors and Affiliations

  1. Institute of Mathematics, Fudan University, Shanghai, 200433, China

    Jixiang Fu

  2. Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA

    Shing-Tung Yau

  3. Shanghai Center for Mathematical Sciences, Shanghai, 200433, China

    Wubin Zhou

Authors
  1. Jixiang Fu
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  2. Shing-Tung Yau
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  3. Wubin Zhou
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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). https://doi.org/10.1007/s00220-015-2337-5

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  • DOI: https://doi.org/10.1007/s00220-015-2337-5

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