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

Valery Alexeev

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.

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

Abstract

We construct a sequence of complete moduli spaces each of which is isomorphic to a weighted projective space. These spaces parameterize certain -dimensional Calabi-Yau varieties associated with the Sylvester sequence . They generalize the moduli space of elliptic curves and Brieskorn's family over , the Baily-Borel compactification of the moduli space of -polarized K3 surfaces. We also study fibrations in such Calabi-Yau varieties, extending to higher dimensions the theory of elliptic surfaces.

Paper Structure

This paper contains 33 sections, 47 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sylvester numbers
  4. Canonical and log canonical singularities
  5. The family $\pi_n$ and its basic properties
  6. Weighted projective space and weighted projective stack
  7. ${\mathbb P}={\mathbb P}(a_0,\dotsc, a_n,1)$
  8. ${\mathcal{P}}(\delta_I)$
  9. ${\mathbb P}(\delta_I)$
  10. ${\mathbb P}(V)$
  11. ${\mathbb P}({\mathcal{V}})$
  12. The family of hypersurfaces
  13. The embedding ${\mathcal{E}}_{n-1}\to {\mathcal{E}}_n$
  14. First properties of the fibers
  15. Singularities of nondegenerate hypersurfaces in ${\mathcal{E}}_n$
  16. ...and 18 more sections

Key Result

Theorem 1

For each $n\ge0$, working over ${\mathbb Z}[1/\prod_{k=0}^n s_k ]$, there is a flat family $\pi_n\colon ({\mathcal{X}}_n,{\mathcal{D}}_n)\to {\mathcal{E}}_n$ of hypersurfaces $X\subset {\mathbb P}$ together with ${\mathbb Q}$-Cartier divisors $D\subset X$ in which every geometric fiber satisfies the The relative dualizing sheaf of $\pi_n$ is $\omega_{{\mathcal{X}}_n/{\mathcal{E}}_n} = \pi_n^*({\ma

Theorems & Definitions (106)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • ...and 96 more