Let $\omega\_{\text{WP}}$ denote the Weil-Petersson metric associated to a family of Calabi-Yau manifolds. That is, let $f : X \to Y$ be a surjective holomorphic map with connected fibres such that, over the regular locus, the fibres $X\_y$ of $f^{\circ} : X^{\circ} \to Y^{\circ}$ are smooth compact Kähler manifolds with $c\_1 =0$ in the $H^2\_{\text{DR}}(X\_y, \mathbb{R})$.


Let $D$ be the associated classifying space, which we know is a symmetric space of non-compact type. Let $\omega\_H$ denote the invariant metric on $D$, which is referred to as the *Hodge metric*. [Lu-Sun](https://arxiv.org/pdf/math/0510021.pdf) showed that the Ricci curvature of $\omega\_{\text{WP}}$ is pinched according to: $$- \omega\_H \leq \text{Ric}(\omega\_{\text{WP}} ) \leq \omega\_H,$$ and $2\omega\_{\text{WP}} \leq \omega\_H$.


*Question:* Is $\text{Ric}(\omega\_{\text{WP}})$ honestly bounded from below, i.e., does there exist a constant $C>0$ such that $$\text{Ric}(\omega\_{\text{WP}}) \geq - C \omega\_{\text{WP}}?$$


I'd be interested in any results concerning the Ricci curvature of the Weil-Petersson metric, even for non-Calabi-Yau families.



Ricci curvature of the Weil-Petersson metric?
A positive answer to this question is given in part (2) of Theorem 3.1. of C.-L. Wang's paper [Curvature properties of the Calabi-Yau moduli](https://www.math.uni-bielefeld.de/documenta/vol-08/18.pdf).


The specific reference is:
Wang, C.-L., *Curvature properties of the Calabi-Yau moduli*, Documenta Math., **8** (2003) 577-590.


