On the geography of $3$-folds via asymptotic behavior of invariants

TL;DR

This paper explores the classification of three-dimensional varieties of general type by analyzing the asymptotic behavior of their invariants, extending techniques from surface geography to higher dimensions.

Contribution

It generalizes methods comparing surface geography with arrangements of curves to threefolds, focusing on asymptotic invariants and singularity resolutions.

Findings

Established asymptotic behavior of invariants for 3-folds.

Analyzed singularities and resolutions in dimension 3.

Extended surface techniques to higher-dimensional varieties.

Abstract

Roughly speaking, the problem of geography asks for the existence of varieties of general type after we fix some invariants. In dimension $1$, where we fix the genus, the geography question is trivial, but already in dimension $2$, it becomes a hard problem in general. In higher dimensions, this problem is essentially wide open. In this paper, we focus on geography in dimension $3$. We generalize the techniques which compare the geography of surfaces with the geography of arrangements of curves via asymptotic constructions. In dimension $2$ this involves resolutions of cyclic quotient singularities and a certain asymptotic behavior of the associated Dedekind sums and continued fractions. We discuss the general situation with emphasis on dimension $3$, analyzing the singularities and various resolutions that show up, and proving results about the asymptotic behavior of the invariants we fix.