On the number and boundedness of log minimal models of general type

TL;DR

This paper establishes bounds on the number of minimal models for varieties and pairs of general type based on volume, Betti numbers, and coefficients, showing boundedness of such models in various settings.

Contribution

It introduces bounds on the number of minimal and log canonical models using volume, Betti numbers, and coefficient conditions, and proves boundedness of families of models.

Findings

Number of minimal models bounded by volume and Betti numbers in dimension 3.

Boundedness of weak log canonical models based on coefficients and volume.

Families of models with fixed volume and coefficient conditions are bounded.

Abstract

We show that the number of marked minimal models of an n-dimensional smooth complex projective variety of general type can be bounded in terms of its volume, and, if n=3, also in terms of its Betti numbers. For an n-dimensional projective klt pair (X,D) with $K_X+D$ big, we show more generally that the number of its weak log canonical models can be bounded in terms of the coefficients of D and the volume of $K_X+D$. We further show that all n-dimensional projective klt pairs (X,D), such that $K_X+D$ is big and nef of fixed volume and such that the coefficients of D are contained in a given DCC set, form a bounded family. It follows that in any dimension, minimal models of general type and bounded volume form a bounded family.