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Let $\mathcal M^{(X,D)}$ be a moduli space of pair varieties $(X,D)$ which $K_X+D$ is trivial and $D$ is a divisor with conic singularities on Kaehler variety $X$. I am looking for a proof that such moduli space exists?

The log Weil-Petersson $\omega_{WP}^D$ correspounding to moduli space $\mathcal M^{(X,D)}$ is Kahler metric?(of course for Weil-Petersson metric it is known, but here we have instead Log Weil-Petersson metric)

A proof or freference for it is appreciated.

Here is some discussion of the moduli space of log Calabi-Yau varieties in case $dim X =2$ in Section 6 of the paper http://arxiv.org/abs/1211.6367

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