Cumrun Vafa with Greene-Shapere-Yau introduced semi-Ricci flat metric here

B. Greene, A. Shapere, C. Vafa, and S.-T. Yau. Stringy cosmic strings and noncompact Calabi-Yau manifolds. Nuclear Physics B, 337(1):1–36, 1990

I know that such metrics are Kahler in fiber direction. Is there any counter example to show that such metrics are not Kahler in horizontal direction?

  • $\begingroup$ This is essentially the Gibbons-Hawking ansatz, why do you expect this is not Kahler? $\endgroup$
    – YHBKJ
    Jun 15, 2015 at 6:13
  • $\begingroup$ This question has been solved in the affirmative by Y-J Choi in arxiv.org/abs/1508.00323 $\endgroup$
    – YangMills
    Aug 9, 2015 at 10:54
  • $\begingroup$ Which page? I couldn't find $\endgroup$
    – user21574
    Aug 30, 2015 at 4:49
  • $\begingroup$ Theorem 1.1 (iii) $\endgroup$
    – YangMills
    Sep 2, 2015 at 20:35
  • 1
    $\begingroup$ That's nice, thanks. This dissertation was announced a while ago in arxiv.org/abs/1201.2930, see footnote on p.7, and I was wondering what happened to it. $\endgroup$
    – YangMills
    Mar 8, 2016 at 4:09

1 Answer 1


A counterexample to the semipositivity of these semi-flat metrics (on elliptic $K3$ surfaces) was constructed by Cao, Guenancia, Paun, Tosatti in this paper, see Theorem 3.1 and the Appendix.


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