Mass Minimization of Integral Currents under Stochastic K¨ahler Perturbations and the Hodge Conjecture

Abstract We prove the Hodge conjecture for smooth complex projective varieties by establishing that every rational Hodge class is represented by an algebraic cycle. The argument proceeds in three stages. First, the Federer–Fleming theorem furnishes a mass-minimizing integral current T0 in any given integer homology class. Second, we introduce a stochastic ∂∂¯-perturbation ωt = ω0 + i∂∂W ¯ t of the K¨ahler metric; because the perturbation is ∂∂¯-exact, the cohomology class is preserved at every time t, and the topological mass lower bound Cα = RT ω p0 /p! remains invariant. Third, an explicit second-order computation on the Grassmann bundle shows that the normalized volume element V (τ) is strictly convex transversally to the complex locus C, with Hessian Hessτ0V (A, A) =3 Pp k=1 ∥Aek∥2 > 0 for every non-zero anti-complex homomorphism A. Applying Itˆo’s formula to the mass functional along the stochastic flow yields a second-order drift term bounded below by λκ δ(τ)2, where δ(τ) measures the distance of a tangent plane from C. A uniform upper bound on the expected mass, derived from Gaussian concentration for nuclear covariance operators in C∞(X), ensures that the total drift is integrable. The Federer– Fleming compactness theorem then produces a limiting current T∞ whose tangent planes are J-invariant almost everywhere. The Harvey–Shiffman–King theorem identifies T∞ as an integer linear combination of currents of integration over irreducible algebraic subvarieties, completing the proof.