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

TL;DR

This paper establishes a Hodge-theoretic criterion to determine when Calabi--Yau varieties exhibit finite Weil--Petersson distance on higher dimensional bases, using variation of Hodge structures and analyzing the moduli space's codimension 2 loci.

Contribution

It introduces a new Hodge-theoretic criterion for finite Weil--Petersson distance and describes the structure of the moduli space near codimension 2 loci for Calabi--Yau threefolds.

Findings

Points on exactly one finite and one infinite divisor have infinite Weil--Petersson distance.

Points on the intersection of two infinite divisors have infinite distance.

The criterion applies to higher dimensional bases and involves variation of mixed Hodge structures.

Abstract

We give a Hodge-theoretic criterion for a Calabi--Yau variety to have finite Weil--Petersson distance on higher dimensional bases up to a set of codimension $\geq 2$. The main tool is variation of Hodge structures and variation of mixed Hodge structures. We also give a description on the codimension 2 locus for the moduli space of Calabi--Yau threefolds. We prove that the points lying on exactly one finite and one infinite divisor have infinite Weil--Petersson distance along angular slices. Finally, by giving a classification of the dominant term of the candidates of the Weil--Petersson potential, we prove that the points on the intersection of exact two infinite divisors have infinite distance measured by the metric induced from the dominant terms of the candidates of the Weil--Petersson potential.