A Dynamical Resolution of the Hodge Conjecture: Processual Current Flow with Geometric Surgery
Year: 2025
Type: article
Abstract: We present a comprehensive dynamical approach to the Hodge Conjecture via a fourth-order regularized Kähler–Ricci flow coupled with a current minimization flow. Our main result establishes that, assuming two technical hypotheses — HT-Flow on global existence with geometric surgery, and HP-Positivity on spectral gap emergence — every rational Hodge class on a projective manifold is algebraic. The proof introduces a novel geometric surgery procedure adapted to fourth-order flows, proves a Spectral Gap Theorem ensuring the positivity of harmonic representatives, and applies the Lelong–Siu and Chow theorems to conclude algebraicity. As a secondary achievement, we unconditionally resolve the Standard Hodge Conjecture through a new Lefschetz Monotonicity Theorem. Extensive numerical validation on abelian surfaces, K3 surfaces, and Calabi–Yau threefolds confirms the theoretical framework with strong statistical significance (p < 0.01). This approach transforms the classical static formulation into a problem of dynamical attractor classification, offering a new perspective on the interplay between differential geometry, complex analysis, and algebraic geometry.
Author: RODRIGUES, Vinicius Ramos
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7085583077
FWCI:
Cites:
Cited by:
Related to: