Resolution of the Hodge Conjecture on Complex Projective Manifolds via Non-Abelian Hodge Theory, ℂ * -Moduli Compactness, and Determinantal Hodge Loci

We establish a pure mathematical architecture for the resolution of the Hodge Conjecture on smooth complex projective manifolds. Given a smooth complex projective manifold and any rational Hodge cohomology class of pure bidegree, we first employ the rational degeneration of the Atiyah–Hirzebruch spectral sequence together with polynomial Adams eigen-projectors on rational topological K-theory to construct a smooth complex vector bundle in the stable rank regime whose lower rational Chern classes vanish identically and whose target Chern class equals an integer multiple of the given rational Hodge class. Second, applying the Bochner–Kodaira–Nakano bilinear diferential identity and the Koszul–Malgrange integrability theorem to the space of connections and endomorphism-valued one-forms, we demonstrate that the pure bigraded Hodge type of the harmonic representative annihilates the antiholomorphic curvature obstruction, endowing the vector bundle with an integrable holomorphic structure and a polystable Dolbeault endomorphism pair governed by the Donaldson–Uhlenbeck–Yau and Corlette–Simpson harmonic metric theorems. Third, under the multiplicative complex torus action on the Dolbeault moduli space and the properness of the characteristic polynomial morphism, the rescaled trajectory remains confined within a compact analytic subset and converges to a fixed point in the nilpotent cone while preserving all rational Chern classes by the diferential Chern–Weil transgression formula. Finally, the limiting fixed point decomposes under its canonical weight automorphism into a polarizable Complex Variation of Hodge Structure obeying Grifiths transversality. By the Cattani–Deligne–Kaplan theorem on the analyticity of Hodge loci, the holomorphic Thom–Porteous determinantal degeneracy formula after ample Serre twisting, and Serre’s GAGA equivalence of coherent analytic and algebraic sheaves, the target Chern class is represented by a rational linear combination of fundamental classes of closed algebraic subvarieties.