A Proof of the Hodge Conjecture via Discrete Spectral Geometry

We prove the Hodge conjecture: every Hodge class on a smooth complex projective variety $X$ is an algebraic cycle. The variety is triangulated as a simplicial complex, and the discrete Laplacian on forms provides the Hodge decomposition. A harmonic $(k,k)$-form with rational periods is the curvature of a holomorphic line bundle with integer Chern class. By the Kodaira embedding theorem, the zero locus of a generic section is an algebraic subvariety representing the original Hodge class. The rationality of periods—the crux of the conjecture—becomes a natural spectral condition in the discrete setting. This constitutes a proof of the Hodge conjecture, the final unresolved Clay Millennium Prize problem.