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

TL;DR

This paper constructs an infinite sequence of complete moduli spaces for Calabi-Yau n-folds, each isomorphic to a weighted projective space, generalizing classical moduli spaces of elliptic curves and K3 surfaces.

Contribution

It introduces a new hierarchy of moduli spaces for Calabi-Yau varieties linked to the Sylvester sequence, extending the theory of elliptic surfaces to higher dimensions.

Findings

Each moduli space is isomorphic to a weighted projective space.

The sequence generalizes classical moduli spaces like elliptic curves and K3 surfaces.

The paper studies fibrations in these Calabi-Yau varieties.

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.