The structure of projective maps between real projective manifolds

TL;DR

This paper investigates the structure of projective maps between compact convex real projective manifolds, revealing finiteness in homotopy classes and manifold structures within each class, especially under strict convexity conditions.

Contribution

It establishes the finiteness of homotopy classes of projective maps and shows each class has a real projective manifold structure, extending understanding of such maps.

Findings

Finite homotopy classes of projective maps

Each homotopy class has a real projective manifold structure

At most one projective map per class when target is strictly convex

Abstract

In this paper we study the set of projective maps between compact proper convex real projective manifolds. We show that this set contains only finitely many distinct homotopy classes and each homotopy class has the structure of a real projective manifold. When the target manifold is strictly convex, our results imply that each non-trivial homotopy class contains at most one projective map. These results are motivated by the theory of holomorphic maps between compact complex manifolds.