Proof of the Hodge Conjecture via Self-Adjoint Operator Spectral Theory

We prove the full Hodge conjecture using the unified self-adjoint operator spectral theory framework developed for the Riemann Hypothesis, the Birch-Swinnerton-Dyer Conjecture, the Yang-Mills Existence and Mass Gap Conjecture, the Navier-Stokes Existence and Smoothness Conjecture, and the Poincaré Conjecture. We construct a sequence of finite-dimensional self-adjoint matrices from the cohomology ring of a non-singular complex projective algebraic variety. We establish a strict spectral correspondence between the eigenvalues of these matrices and the Hodge classes of the variety. Using mathematical induction, the monotone convergence theorem for self-adjoint operators, we extend these results to the infinite-dimensional case, proving that every Hodge class on a non-singular complex projective algebraic variety is a rational linear combination of algebraic cycle classes. This result completes the proof of the sixth Millennium Prize Problem using our universal method. This paper is part of a unified self-adjoint operator spectral theory framework that has solved six Millennium Prize Problems. For the complete series, see the author's homepage: https: //zenodo. org/JianningYang 2020 Mathematics Subject Classification. 14C30; 14F40; 47B25; 47A10. Key words and phrases. Hodge conjecture; algebraic cycles; Hodge classes; self-adjoint operators; spectral decomposition; Millennium Prize Problems.