Topology of convex projective four-manifolds

I will discuss topological obstructions for a closed four-manifold that admits a (real) properly convex projective structure. I will present a vanishing result on the Pontryagin classes, that in dimension four provides information on the geography problem for properly convex projective manifolds, and implies the solution to a question of Yves Benoist on the (in)compatibility of real convex projective structures and complex hyperbolic structures. Time permitting, I will also discuss the construction of examples that realize all possible positive values of the Euler characteristic. Based on joint work with Stefano Riolo and Leone Slavich.