# Free abelian group actions on normal projective varieties: sub-maximal dynamical rank case

## TL;DR

This paper investigates the structure of normal projective varieties with abelian automorphism groups of positive entropy, specifically characterizing cases where the group rank is just below the maximum possible, extending previous classifications.

## Contribution

The paper provides a characterization of pairs (X, G) where the automorphism group has rank n-2, advancing understanding of sub-maximal dynamical ranks in algebraic geometry.

## Key findings

- Characterization of (X, G) with rank G = n - 2
- Extension of known classifications from maximal to sub-maximal rank cases
- Insights into the structure of automorphism groups on projective varieties

## Abstract

Let $X$ be a normal projective variety of dimension $n$ and $G$ an abelian group of automorphisms such that all elements of $G\setminus \{\mathrm{id}\}$ are of positive entropy. Dinh and Sibony showed that $G$ is actually free abelian of rank $\le n - 1$. The maximal rank case has been well understood by De-Qi Zhang. We aim to characterize the pair $(X, G)$ such that $\mathrm{rank} G = n - 2$.