From Energy Alone: Energetic–Geometric Realization of Rational Hodge Classes via Scalar Curvature Forms

This article develops an energetic–geometric framework for realizing rational Hodge classes via scalar curvature fields on compact Kähler (in particular, complex projective) varieties. We introduce a smooth energetic potential EEE and the associated elliptic operator ΔE\Delta_EΔE whose principal symbol agrees with the Hodge Laplacian, define harmonic projection PEP_EPE that preserves Hodge type (p,p)(p,p)(p,p), and construct differential forms of type (p,p)(p,p)(p,p) as energetic curvature expressions. We prove: (i) harmonic approximation of rational (p,p)(p,p)(p,p)-classes by PEP_EPE-harmonic energetic forms; (ii) an analytic localization lemma yielding support control in tubular neighborhoods; and (iii) Thom/Gysin compatibilities that identify the resulting cohomology class with that of an algebraic cycle (modulo torsion) under stated hypotheses. Together these results show that every rational Hodge class admits an energetic curvature representative cohomologous to the class and supported on an algebraic cycle. Quantitative heat-kernel and domain estimates required by (i)–(iii) are established in the companion technical part.