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?
> 
> 
> 



Weil-Petersson metric is quasi isometric with which model?
Ken-Ichi Yoshikawa, found an asymptotic formula for Weil-Petersson metric on moduli space of Calabi-Yau varieties as follows


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


