Calibration–Coercivity and the Hodge Conjecture: A Fully Classical, Quantitative Reduction with c = 1/3

Abstract

We present a purely classical, quantitative route to the Hodge conclusion for rational (p,p) classes on smooth projective Kähler manifolds. Let E(α) denote the Dirichlet energy of a smooth closed 2p-form α representing a fixed (p,p) cohomology class, and let γ_harm be the ω-harmonic representative of that class. We define a geometric distance-to-calibration functional measuring the L2 distance to the convex calibrated cone K_p associated to the Kähler calibration φ = ω^p/p! and denote it by Def_cone(α). We prove the cone-based calibration–coercivity inequality E(α) − E(γ_harm) ≥ c · Def_cone(α) with c = 1/3. Consequently, any minimizing sequence of closed representatives in a rational (p,p) class has vanishing calibration defect and converges (in the weak sense of currents) to a positive, calibrated (p,p) current that saturates the Wirtinger bound, hence is the current of integration over a complex analytic p‑cycle; projectivity upgrades it to an algebraic cycle. This yields the Hodge conclusion for the class. The argument is intrinsic to Kähler geometry and isolates a clean, verifiable inequality with fully explicit constants.