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

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.