Euler characteristic and flat cohomology of convex projective $4$-manifolds

Let $M=Ω/Γ$ be a closed properly convex real projective $4$-manifold. We prove the Chern-Hopf-Thurston conjecture for this class of manifolds, i.e. $χ(M)\geq0$. The argument transfers Akagawa's double-form identity on the Cheng--Yau cone over $Ω$ to a Bochner identity on $M$, from which we deduce th