Existence of the equivariant minimal model program for compact K\"ahler threefolds with the action of an abelian group of maximal rank

TL;DR

This paper proves that for certain compact K"ahler threefolds with a maximal rank abelian group action, the minimal model program results in the variety being either rationally connected or a complex torus, fixing previous proof issues.

Contribution

It establishes the existence of the equivariant minimal model program for these threefolds and clarifies a previous proof gap.

Findings

Varieties are either rationally connected or bimeromorphic to a complex torus.

The equivariant minimal model program can be successfully run in this setting.

Addresses and corrects a previous proof issue.

Abstract

Let $X$ be a $\mathbb{Q}$-factorial compact K\"ahler klt threefold admitting an action of a free abelian group $G$, which is of positive entropy and of maximal rank. After running the $G$-equivariant log minimal model program, we show that such $X$ is either rationally connected or bimeromorphic to a $Q$-complex torus. In particular, we fix an issue in the proof of our previous paper.