Mathematical Research Letters
Volume 25 (2018)
A Hodge theoretic criterion for finite Weil–Petersson degenerations over a higher dimensional base
Pages: 617 – 647
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.
I am grateful to Professor C.-L. Wang for introducing this problem to me and for sharing with me his viewpoints. I am grateful to him and to Professor H.-W. Lin for their steady encouragement. I am also grateful to Professors Z. Lu and J.-D. Yu, and Dr. T.-C. Yang for discussions. Results in this paper had been presented in the conference Calabi–Yau Geometry and Mirror Symmetry in 2014 at National Taiwan University. I am grateful to Professor S.-T. Yau for the invitation and for his interest in this work.
Received 28 April 2016
Accepted 16 January 2017
Published 5 July 2018