Curvature–Cohomology Criterion for Projectivity: A Synthesis of Classical Results in Hodge Theory

This paper synthesizes classical results in Hodge theory, curvature positivity, and vanishing theorems to give a concise curvature–cohomology criterion for the projectivity of compact Kähler manifolds. While each analytic component—Yau’s solution of the Calabi conjecture, the Bochner–Kodaira–Nakano identity, and Kodaira’s embedding theorem—is well-known, their combination yields a transparent geometric criterion: if the first Chern class c1(M) admits a semi-positive real (1,1) representative that is strictly positive at some point (or equivalently has a maximal rank n somewhere), then M is projective. Beyond the maximal rank case, we refine Girbau’s classical vanishing theorem to obtain an optimal rank-sensitive bound: if 2πc1(M) has a semi-positive representative whose pointwise rank is k somewhere, then Hp,0(M)=0 for all p>n−k. This sharpens the classical Girbau–Griffiths–Harris vanishing theorem and quantifies how partial positivity of a Ricci representative constrains Hodge cohomology. We situate these criteria alongside classical tests (Kodaira integrality and Moishezon) and numerical descriptions of the Kähler cone (Demailly–Paun), discuss deformation-invariance properties, and relate them to RC positivity and Campana–Peternell-type statements. Examples illustrate the sharpness of the hypotheses, and we survey the effective bounds—ranging from rigorous uniform high ampleness results to conjectural optimal constants—with clear distinction between proven theorems, refinements of classical results, and open problems. The contribution of this work lies not in new analytic techniques but in (1) isolating a sharp curvature condition at the level of c1(M); (2) organizing classical tools into a direct projectivity criterion; and (3) clarifying the rank-dependent vanishing behavior that follows from partial positivity.