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