A tower of complete moduli spaces of Calabi-Yau $n$-folds

Abstract

We construct a sequence of complete moduli spaces $$E_0 \subset E_1 \subset E_2 \subset \dots E_n \subset\dots,$$ each of which is isomorphic to a weighted projective space. These spaces parameterize certain $n$-dimensional Calabi-Yau varieties associated with the Sylvester sequence $2,3,7,43,\dots$. They generalize the moduli space of elliptic curves $\overline{M}_{1,1}=\mathbb P(4,6)$ and Brieskorn's family over $\overline{F}^{\rm BB}_{U\oplus E_8} = \mathbb P(4,10,\dotsc, 42)$, the Baily-Borel compactification of the moduli space of $U\oplus E_8$-polarized K3 surfaces. We also study fibrations in such Calabi-Yau varieties, extending to higher dimensions the theory of elliptic surfaces.