On existence of log minimal models II

TL;DR

This paper establishes that the existence of log minimal models in a given dimension implies the LMMP with scaling, and that a weak nonvanishing conjecture leads to the minimal model conjecture in the same dimension.

Contribution

It demonstrates the logical implications between key conjectures and the existence of log minimal models within the minimal model program.

Findings

Existence of log minimal models implies LMMP with scaling.

Weak nonvanishing conjecture implies minimal model conjecture.

Results connect conjectures in the minimal model program.

Abstract

We prove that the existence of log minimal models in dimension $d$ essentially implies the LMMP with scaling in dimension $d$. As a consequence we prove that a weak nonvanishing conjecture in dimension $d$ implies the minimal model conjecture in dimension $d$.