I would like to know which explicit metrics on noncompact Calabi-Yau (CY) threefolds are known. For instance, an important class of such spaces can be constructed algebraically, including local $\mathbb{CP}^1$ (a.k.a. the resolved conifold), local $\mathbb{CP}^2$, local $\mathbb{CP}^1\times \mathbb{CP}^1$, and the deformed conifold. However, as far as I have searched the math and physics literature, I have found explicit CY metrics only in the case of the resolved and deformed conifold. Is the CY metric for, e.g., local $\mathbb{CP}^2$ known? What about other cases?

  • $\begingroup$ Welcome to Math.SE! I'm not familiar with your terminology ("local $\mathbf{CP}^{n}$", "conifold", etc.). Could you perhaps provide references for the metrics you mention? Also, is your definition of "Calabi-Yau metric" a complete, Ricci-flat Kähler metric? If so, the earliest reference that comes immediately to mind is Calabi's Métriques kählériennes et fibrés holomorphes. $\endgroup$ – Andrew D. Hwang Jul 22 '15 at 14:18
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    $\begingroup$ Hi Andrew, Thank you for your response! I'm following terminology which is standard in some physics circles. Specifically, in my post I'm following the discussion in arxiv.org/abs/hep-th/0410178. Regarding your question, yes I mean complete Ricci-flat Kahler metrics. $\endgroup$ – Nuno Jul 22 '15 at 14:20
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    $\begingroup$ Thank you for the reference/clarification! If you'll excuse the self promotion, A momentum construction for circle-invariant Kähler metrics may be worth a look. Particularly, the metrics Neitzke and Vafa call local $\mathbf{CP}^{2}$, local $\mathbf{CP}^{1} \times \mathbf{CP}^{1}$, and resolved conifold do have explicit constructions, in the sense that the metric can be constructed from a single explicit rational function of one variable. Are you asking just out of curiosity, or would the details be helpful to your work? $\endgroup$ – Andrew D. Hwang Jul 22 '15 at 15:49

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