An Optimization-Theoretic Framework for the Hodge Conjecture: Harmonic Projection and Algebraic Cycle Approximation

The Hodge conjecture is one of the most profound unsolved problems in algebraic geometry and topology. It asserts that on a complex projective manifold, every rational Hodge class can be represented as a rational linear combination of algebraic cycles. Traditional research relies primarily on algebraic geometry, homology theory, and Kähler geometric structures. This paper proposes a new research paradigm: transforming the representation problem of Hodge classes into a global optimization problem and constructing a harmonic projection optimization framework. We introduce a new optimization functional, the Hodge Energy Functional, and propose a novel dynamical system, the Hodge Projection Flow. This method equivalently transforms the Hodge conjecture into an energy minimization problem and proposes a new equation, the Hodge Optimization Equation. This theoretical framework unifies Hodge decomposition, variational analysis, and geometric optimization, providing a new mathematical path for studying the Hodge conjecture.