Convex projective manifolds, symmetric spaces and geometric decompositions

We prove that if a closed, indecomposable, properly convex real projective $4$-manifold is geometric or admits a geometric decomposition in the sense of Thurston, then every piece is real hyperbolic.

This extends a theorem of Benoist to dimension four.

Moreover, we build orientable (non-hyperbolic) $4$-manifolds of the above type, with arbitrary positive, even, Euler characteristic.

Along the way, we characterise the compact locally symmetric spaces that virtually support properly convex real projective structures.