Force wider viewport size in iOS Safari?
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
$$\omega_{WP}=\int_{X/Y}\rho^{n+1}$$