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](https://en.wikipedia.org/wiki/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>



Canonical metric on moduli space of singular Calabi-Yau varieties
This question has been solved in the paper of [Y.Odaka,](http://annals.math.princeton.edu/2013/177-2/p06) 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](http://www.math.jhu.edu/~yuan/krfcy.pdf) 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](http://archiv.ub.uni-marburg.de/diss/z2015/0401/pdf/dmb.pdf) PhD thesis


$$\omega\_{WP}=\int\_{X/Y}\rho^{n+1}$$


