On existence of log minimal models

TL;DR

This paper establishes that the log minimal model program in lower dimensions implies the existence of log minimal models in higher dimensions, specifically proving their existence in dimension five and providing new proofs in dimension four.

Contribution

It shows that the log minimal model program in dimension d-1 ensures the existence of models in dimension d, extending known results and simplifying proofs.

Findings

Log minimal models exist for effective lc pairs in dimension five.

New proofs for the existence of models in dimension four.

Connections between lower-dimensional programs and higher-dimensional models.

Abstract

In this paper, we prove that the log minimal model program in dimension $d-1$ implies the existence of log minimal models for effective lc pairs (eg of nonnegative Kodaira dimension) in dimension $d$. In fact, we prove that the same conclusion follows from a weaker assumption, namely, the log minimal model program with scaling in dimension $d-1$. This enables us to prove that effective lc pairs in dimension five have log minimal models. We also give new proofs of the existence of log minimal models for effective lc pairs in dimension four and the Shokurov reduction theorem. Other applications appear in a paper of Birkar-Paun.