Smooth Calabi-Yau varieties with large index and Betti numbers

TL;DR

This paper constructs high-dimensional smooth Calabi-Yau varieties with doubly exponential growth in index and Betti numbers, conjecturing their extremality and extending known examples to higher dimensions.

Contribution

It introduces new constructions of smooth Calabi-Yau varieties with maximal index and Betti numbers, surpassing previously known examples in small dimensions.

Findings

Constructed smooth Calabi-Yau varieties with doubly exponential index in every dimension.

Produced varieties with extremal topological invariants, including Euler characteristics and Betti sums.

Conjectured these constructions to be maximal and extremal in all dimensions.

Abstract

A normal variety $X$ is called Calabi-Yau if $K_X \sim_{\mathbb Q} 0$. The index of $X$ is the smallest positive integer $m$ so that $m K_X \sim 0$. We construct smooth, projective Calabi-Yau varieties in every dimension with doubly exponentially growing index, which we conjecture to be maximal in every dimension. We also construct smooth, projective Calabi-Yau varieties with extreme topological invariants; namely, their Euler characteristics and the sums of their Betti numbers grow doubly exponentially. These are conjecturally extremal in every dimension. The varieties we construct are known in small dimensions but we believe them to be new in general. This work builds off of the singular Calabi-Yau varieties found by Esser, Totaro, and Wang in arXiv:2209.04597.