On the Incompleteness of the Weil-petersson Metric along Degenerations of Calabi-yau Manifolds

The classical Weil-Petersson metric on the Teichmüller space of compact Riemann surfaces is a Kähler metric, which is complete only in the case of elliptic curves [Wo]. It has a natural generalization to the deformation spaces of higher dimansional polarized Kähler-Einstein manifolds. It is still Kähler, and in the case of abelian varieties and K3 surfaces, the Weil-Petersson metric turns out to coincide with the Bergman metric of the Hermitian symmetric period domain, hence is in fact “complete” Kähler-Einstein [Sc]. The completeness is an important property for differential geometric reason. Motivated by the above examples, one may naively think that the completeness of the Weil-Petersson metric still holds true for general Calabi-Yau manifolds (compact Kähler manifolds with trivial canonical bundle). However, explicit calculation done by physicists (eg. Candelas et al. [Ca] for some special nodal degenerations of Calabi-Yau 3-folds) indicated that this may not always be the case. The notion of completeness depends on the precise definition the “moduli space”. However, through our analysis, it would become clear that the Weil-Petersson metric is in general incomplete if one sticks