A Hodge theoretic criterion for finite Weil-Petersson degenerations over a higher dimensional base (Q1990273)

This paper addresses the validity of the following conjecture due to \textit{C.-L. Wang} [Math. Res. Lett. 4, No. 1, 157--171 (1997; Zbl 0881.32017)]. \par Conjecture. Let ${\mathfrak{X}}/S$ be an $n$-dimensional Calabi-Yau degeneration which is smooth outside a simple normal crossing divisor $\bigcup_i E_i$. Then $X:={\mathfrak{X}}_s$ with $s\in\bigcup_i E_i$, has finite Weil-Petersson distance if and only if $N_iF^n_{\infty}(s)=0$ for all $i$ with $s\in E_i$. Here $F^n_{\infty}$ us the $n$-th piece of Schmid's limiting Hodge filtration and $N_i$ is the nilpotent part of monodromy around $E_i$. \par The main results of this article are formulated as follows. \par Theorem 1: The conjecture holds for the degeneration of a Calabi-Yau $n$-folds, up to a set of codimension $\geq 2$ in the base $S$. \par Proof uses variation of mixed Hodge structures on codimension one boundary points. This imposes a strong constraint on the Weil-Petersson potential and hence the Weil-Petersson metric can be controlled. \par The Weil-Petersson metric on Calabi-Yau threefolds is computed on the covering ${\mathbb{H}}^2\times(\Delta)^{r-2}\to (\Delta^*)^2\times (\Delta)^{r-2}$, where $\Delta:=\{\, z\in{\mathbb{C}}\,|\, |z|<1\,\}$, $\Delta^*:=\Delta\setminus \{0\}$. \par Theorem 2: Let ${\mathfrak{X}}/S$ be a degeneration of Calabi-Yau threefolds as in the conjecture. Suppose that $s\in S$ lies on exactly two boundary divisors, say $s\in E_1\cap E_2$, with $E_1$ infinite and $E_2$ finite. Then $s$ has infinite Weil-Petersson distance along the angular slices $\{\,(z_1,z_2)\in {\mathbb{H}}^2\,|\, \text{Re}(z_j)=c_j\in{\mathbb{R}}\,\}\times(\Delta)^{r-2}.$ \par The next result handles the case when $s$ lies in the intersection of two infinite divisors. \par Theorem 3: In the case of Calabi-Yau threefolds, suppose $s\in S$ lies exactly on two boundary divisors, say $s\in E_1\cap E_2$ with both $E_i$ infinite. Then it has infinite distance measured by the dominant term of the candidates of the Weil-Petersson potentials.