Radial Coercivity and the Chern–Hopf–Thurston Conjecture for Convex Projective Manifolds

Let M be a closed oriented properly convex real projective manifold of dimension d, and let E be its rank-N = d+1 standard flat bundle associated with cone-preserving unimodular holonomy. Starting from Tamburelli's Bochner identity, we use radial contraction to obtain a quantitative lower bound for a nonnegative term and thereby establish the lower bound Δp,q ≥ (p−q)(N−p−q)/N for the Hodge Laplacian on exterior-power flat-bundle-valued forms. Consequently, the cohomology of the p-th exterior-power dual bundle is supported in the interval from min(p,N−p) to d−min(p,N−p). In even dimension d = 2n, this gives the identity dim Hⁿ(M; ΛⁿE*) = binomial(2n+1,n)(−1)ⁿχ(M) ≥ 0. In odd dimension, a central exterior-power local system is acyclic. Using published sofic rank-twisting and complex-rank approximation theorems, we also obtain ordinary Singer vanishing for this class of manifolds. For nonorientable manifolds, the Euler-sign and Singer conclusions descend from the orientation double cover. The geometric reduction and the four-dimensional sign theorem are due to Tamburelli; the present argument refines his arbitrary-dimensional operator identity.