Rigidity of complex convex divisible sets

TL;DR

This paper investigates complex convex divisible sets in projective space, proving that the only such sets with smooth boundary are projective balls, thus extending convexity rigidity results from real to complex settings.

Contribution

It establishes a rigidity theorem for complex convex divisible sets, showing they must be projective balls if they have a smooth boundary.

Findings

Only projective balls are divisible complex convex sets with C^1 boundary.

Extends convexity rigidity from real to complex projective spaces.

Provides new insights into the structure of complex convex divisible sets.

Abstract

An open convex set in real projective space is called divisible if there exists a discrete group of projective automorphisms which acts co-compactly. There are many examples of such sets and a theorem of Benoist implies that many of these examples are strictly convex, have $C^1$ boundary, and have word hyperbolic dividing group. In this paper we study a notion of convexity in complex projective space and show that the only divisible complex convex sets with $C^1$ boundary are the projective balls.