A Solution to the Hodge Conjecture by C.H.F Warnke

This work proposes a variational approach to the Hodge Conjecture for smooth, projective complex Kähler manifolds. The author introduces a geometrically motivated functional Ω[α], defined on the space of harmonic representatives of rational Hodge classes α ∈ H²ᵖ(X,ℚ) ∩ Hᵖ,ᵖ(X). It is proven that minimizing this functional precisely selects those classes that are representable by algebraic cycles. The method draws on classical Hodge theory, spectral decompositions of harmonic forms, and projective approximations in cohomology. The paper includes both an intuitive, conceptual exposition and a formal mathematical treatment, complete with definitions, lemmas, and theorems. Applications to K3 surfaces, projective products, and Calabi–Yau varieties illustrate the power of the technique. This framework offers a novel, functionally analytic perspective on one of algebraic geometry’s central open problems.