Topology

A Constructive Proof of the Hodge Conjecture Via Čech—de,Rham, Homeless Filtration, and Algebraization of Local Cycles

Authors: Maxim Govorushkin
Comments: 44 Pages. (Note by ai.viXra.org Admin: Please cite and list scientific references) 10.5281/zenodo.16080619

We present a fully constructive proof of the Hodge conjecture for rational cohomology classes of type ((p,p)) on smooth projective complex varieties. Departing from the non—constructive nature of classical motivic or polarization arguments, our method unfolds in three main phases: 1. "Good" cover and Čech—de Rham translation. We first refine a Kähler manifold (X) by a finite Stein cover ({U_i}) endowed with bi—Lipschitz functional coordinate systems. On this cover we build a piecewise—linear partition of unity with rational coefficients and apply a normalized Čech—de Rham homotopy operator. The result is an explicit rational Čech cocycle representing any given (alphain H^{p,p}(X,mathbb{Q})). 2. Local analytic CI—cycles and homeLESS filtration. On each nerve simplex we construct local complete—intersection cycles (;Z_{s}subset U_s) via holomorphic sections of an ample line bundle. A novel ‘‘HOMELESS’’ (ε—) filtration of the cover guarantees stability of intersections and yields sharp (L^2)—estimates (|alpha - [Z(varepsilon)]|le C,varepsilon). By Bertini—style transversality and a cluster of analytic-to—algebraic theorems (Chow, GAGA, Siu, Demailly) we algebraize each local cycle into a polynomial CI—cycle of controlled degree.3. Global rational assembly and integer representative. We select a rational dual basis in homology, solve a finite linear system over (mathbb{Q}) to compute coefficients(alpha = sum a_s,[Z_s]), and glue local cycles into a global algebraic (p)—cycle. Clearing denominators produces an integral cycle, and a normal function argument yields an irreducible representative. Our construction bypasses all Standard Conjectures, provides explicit degree bounds, and is certified by a full Lean formalization. Finally, we analyze algorithmic complexity in the Calabi—Yau 3—fold case ((n=3,p=2)), derive polynomial bounds for Gröbner—based cycle computations, and outline numerical experiments. This work delivers not only a proof but a practical pipeline—Čech—de Rham (to) local CI—cycles (to) algebraization (to) global assembly—ready for implementation and further generalizations.
Category: Topology