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Let $\pi:X\to Y$ be a surjective holomorphic map with connected fibers and let fibers are singular Calabi-Yau varieties (i.e. numerical dimension is zero) then is it possible to construct canonical metric like Weil-Petersson metric on moduli space of such fibers which centeral fiber $X_0$ has not mild singularity (canonical singularity)?

Motivation: In fact when centeral fiber $X_0$ has only canonical singularity then Ken-Ichi Yoshikawa showed that the Weil-Petersson metric is bounded by blow up Poincare model metric. I want to see what happen if we don't have such assumption see http://arxiv.org/pdf/1007.2836.pdf

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  • $\begingroup$ Could you please clarify what you mean by 'degeneration of Calabi-Yau varieties with singular fibers'? $\endgroup$ – Piotr Achinger Apr 8 '16 at 22:01
  • $\begingroup$ I don't understand the initial sentence singular "Calabi-Yau varieties (i.e. numerical dimension is zero)". Do you mean to say that if I take a complex 2-torus and blow up a point then the resulting manifold is Calabi-Yau, for you? $\endgroup$ – YangMills Apr 12 '16 at 22:50
  • $\begingroup$ I have assumed fibers are not smooth. numerical dimension is just numerical Kodaira dimension $\endgroup$ – user21574 Apr 13 '16 at 0:20
  • $\begingroup$ Here ms.u-tokyo.ac.jp/journal/pdf/jms220115.pdf Takayama for projective Calabi-Yau fibration showed that when centeral fiber has at worst canonical singularities then diameter of ibres are bounded and Weil-Petersson metric has upper bounded potential $\endgroup$ – user21574 Jun 28 '16 at 19:43
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This question has been solved in the paper of Y.Odaka, saying that a polarized Calabi-Yau have at worst canonical singularities since they are K-stable. So if the central fibre $X_0$ be Calabi-Yau variety with canonica singularities then all general fibers $X_t$ are Calabi-Yau varieties with at worst canonical singularities and by recent result of Song-Yuan there exists a Ricci flat metric on each fibre and we can introduce fiberwise Ricci flat metric $\rho$ (introduced by Greene-Shapere-Vafa-Yau, see Brian R. Greene, Alfred Shapere, Cumrun Vafa, and Shing-Tung Yau, Stringy cosmic strings and noncompact Calabi-Yau manifolds, Nuclear Phys. B 337 (1990))

Hence we can introduce Weil-Petersson metric by fiber integral as follows see Matthias Braun's PhD thesis

$$\omega_{WP}=\int_{X/Y}\rho^{n+1}$$

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