Overview: This work introduces an autonomous analytic approach to the Hodge Conjecture. Instead of searching for algebraic cycles directly, the research focuses on selecting an equilibrium Kähler metric through a coupled Monge–Ampère system with moment constraints (MA+M).
Key Innovation: We establish the Harmonic Orthogonality Condition (HOC) as a necessary condition for algebraicity. The proof shows that if an MA+M metric satisfying HOC exists, the harmonic representative is structurally forced to align with the Harvey–Lawson calibration, ensuring that the mass minimizer in that class represents an algebraic cycle.
The Hodge Conjecture: Monge–Ampère with Moment Constraints
Abstract: We introduce an autonomous analytic framework for the Hodge Conjecture based on the selection of an equilibrium metric within the Kähler class. The approach centers on a coupled Monge–Ampère system with moment constraints (MA+M) and the formulation of the Harmonic Orthogonality Condition (HOC), which requires the vanishing of the primitive component of the harmonic representative. The existence of a metric satisfying HOC is established as a necessary condition for the algebraicity of Hodge classes: such a metric forces the mass minimizer in a given (k,k)-class to be calibrated by the Harvey–Lawson calibration. The proof integrates classical Yau–Calabi–Aubin theory with the spectral stability of Hodge projections and the identification of the mass subgradient with algebraic cycles
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