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arXiv · 2610.31085

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

Andrea Tamburelli
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Abstract

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 that $H^q(M;Λ^pE^*)=0$ 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 $Λ^2E^*$ concentrates in degree two.

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Andrea Tamburelli. 2026-10-22. Euler characteristic and flat cohomology of convex projective $4$-manifolds. https://arxiv.org/abs/2610.31085

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