  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.
