Conjecture of Tits type for complex varieties and Theorem of Lie-Kolchin type for a cone

TL;DR

This paper formulates and proves a Lie-Kolchin type theorem for cones, proposes a Tits type conjecture for automorphism groups of complex varieties, and verifies it for several classes of complex manifolds.

Contribution

It introduces a new Lie-Kolchin type theorem for cones and confirms a Tits type conjecture for various complex varieties, extending understanding of automorphism groups.

Findings

Proved Lie-Kolchin type theorem for cones.

Formulated and verified Tits type conjecture for complex tori, hyperk"ahler manifolds, surfaces, and threefolds.

Derived algebro-geometric consequences from the theorems.

Abstract

First, we shall formulate and prove Theorem of Lie-Kolchin type for a cone and derive some algebro-geometric consequences. Next, inspired by a recent result of Dinh and Sibony we pose a conjecture of Tits type for a group of automorphisms of a complex variety and verify its weaker version. Finally, applying Theorem of Lie-Kolchin type for a cone, we shall confirm the conjecture of Tits type for complex tori, hyperk\"ahler manifolds, surfaces, and minimal threefolds.