Holomorphic actions, Kummer examples, and Zimmer Program

TL;DR

This paper classifies certain compact Kähler manifolds with lattice actions from high-rank Lie groups, advancing the Zimmer program and characterizing specific complex tori through their automorphism groups.

Contribution

It introduces a new classification of Kähler manifolds under high-rank Lie group actions, offering insights into the Zimmer program and automorphism group properties.

Findings

Classification of compact Kähler manifolds with lattice actions

Characterization of complex tori via automorphism groups

Advancement in the Zimmer program

Abstract

We classify compact K\"ahler manifolds $M$ of dimension $n\geq 3$ on which acts a lattice of an almost simple real Lie group of rank $\geq n-1$. This provides a new line in the so-called Zimmer program, and characterizes certain type of complex tori by a property of their automorphisms groups.