Varieties of general type with doubly exponential asymptotics

TL;DR

This paper constructs examples of high-dimensional varieties of general type with minimal volume and maximal vanishing plurigenera, demonstrating doubly exponential decay rates in volume as dimension increases.

Contribution

It provides the first known constructions achieving doubly exponential volume decay in high dimensions for varieties of general type and related classes.

Findings

Constructed varieties with minimal volume in high dimensions.

Achieved doubly exponential decay rate of volume with dimension.

Examples exhibit doubly exponential behavior in various types of varieties.

Abstract

We construct smooth projective varieties of general type with the smallest known volume and others with the most known vanishing plurigenera in high dimensions. The optimal volume bound is expected to decay doubly exponentially with dimension, and our examples achieve this decay rate. We also consider the analogous questions for other types of varieties. For example, in every dimension we conjecture the terminal Fano variety of minimal volume, and the canonical Calabi-Yau variety of minimal volume. In each case, our examples exhibit doubly exponential behavior.