Automorphisms of projective manifolds

TL;DR

This paper investigates the topological structure of compact projective manifolds with non-discrete automorphism groups, classifies certain 3-dimensional cases, and links automorphism properties to radiant affine manifolds.

Contribution

It introduces a method to analyze the topological properties of projective manifolds via automorphism groups and classifies 3D cases with large automorphism groups.

Findings

Orbits of the connected automorphism group are immersed projective submanifolds.

Classified 3D compact projective manifolds with automorphism dimension at least 2.

Connected automorphism group orbits have specific topological properties.

Abstract

Let $(M,P\nabla_M)$ be a compact projective manifold and $Aut(M,P\nabla_M)$ its group of automorphisms. The purpose of this paper is to study the topological properties of $(M,P\nabla_M)$ if $Aut(M,P\nabla_M))$ is not discrete by applying the results that I have shown in [13] and the Benzekri's functor which associates to a projective manifold a radiant affine manifold. This enables us to show that the orbits of the connected component of $Aut(M,P\nabla_M)$ are immersed projective submanifolds. We also classify $3$-dimensional compact projective manifolds such that $dim(Aut(M,P\nabla_M))\geq 2$.