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

Abstract

Let $M=\Omega/\Gamma$ be a closed properly convex real projective $4$-manifold. We prove the Chern-Hopf-Thurston conjecture for this class of manifolds, i.e. $\chi(M)\geq0$. The argument transfers Akagawa's double-form identity on the Cheng--Yau cone over $\Omega$ to a Bochner identity on $M$, from which we deduce that $H^q(M;\Lambda^pE^*)=0$ whenever $q<p(n+1-p)/(n+1)$, in arbitrary dimension $n$, for some flat bundle $E$ naturally attached to $M$. In dimension four, this implies that the cohomology with coefficients in $\Lambda^2E^*$ concentrates in degree two.