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Let $\mathcal M_g$ be the moduli space of curves of genus $g$. If we take $X^{reg}=X\setminus D$, where D is a divisor with normal crossings. Endow $X^{reg}$ with a complete Kahler metric which has a type of singularities normal to each component of $D$; in local coordinates, if $D = (z_1,...,z_k)$, the Weil-Petersson metric on moduli space of $\mathcal M_g$ is quasi-isometric with the following model, $$\sum_{i=1}^k\frac{dz_i\wedge d\bar z_i}{|z_i|^2(-\log |z_i|)^3}+\sum_{i=k+1}^n dz_i\wedge d\bar z_i$$

See corolary 4, $L^2$-cohomology of the Weil Petersson metric. Mapping class groups and moduli spaces of Riemann surfaces (Göttingen, 1991/Seattle, WA, 1991), 345–360, Contemp. Math., 150, Amer. Math. Soc., Providence, RI, 1993.

Let $\omega_{WP}$ be the Weil-Petersson metric on moduli space of Calabi-Yau varieties, then $\omega_{WP}$ is quasi-isometric with which model?

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Ken-Ichi Yoshikawa, found an asymptotic formula for Weil-Petersson metric on moduli space of Calabi-Yau varieties as follows

$$\omega_{WP}=\left\{\frac{\ell }{|s|^2(\log |s|)^2}+O\left(\frac{1 }{|s|^2(\log |s|)^3}\right)\right\}\sqrt[]{-1}ds\wedge d\bar s$$

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