Calabi-Yau varieties of large index

TL;DR

This paper constructs high-dimensional Calabi-Yau varieties with the largest known indices, which grow doubly exponentially with dimension, and provides evidence supporting their maximality.

Contribution

The authors construct Calabi-Yau varieties with the largest known indices in high dimensions, using mirror symmetry and providing supporting evidence for their maximality.

Findings

Index grows doubly exponentially with dimension

Constructed examples have the largest known index

Supports conjecture of maximal index in low dimensions

Abstract

Call a projective variety $X$ Calabi-Yau if its canonical divisor is ${\bf Q}$-linearly equivalent to zero. The smallest positive integer $m$ with $mK_X$ linearly equivalent to zero is called the index of $X$. We construct Calabi-Yau varieties with the largest known index in high dimensions. In our examples, the index grows doubly exponentially with dimension. We conjecture that our examples have the largest possible index, with supporting evidence in low dimensions. The examples are obtained by mirror symmetry from our Calabi-Yau varieties with an ample Weil divisor of small volume. We also give examples for several related problems, including Calabi-Yau varieties with large orbifold Betti numbers or small minimal log discrepancy.