Geometry

2507 Submissions

[2] ai.viXra.org:2507.0098 [pdf] submitted on 2025-07-21 17:52:52

Hodge Conjecture as a Theorem of Functional Geometry

Authors: Maxim Govorushkin
Comments: 72 Pages. 10.5281/zenodo.16279209

In this paper we develop a fully constructive proof of the Hodge Conjecture for rational classes of type ((p,p)) on a smooth projective complex manifold (M). Rather than relying on deep motivic or trans-cendental arguments, we introduce a new emph{functional—geometric} (FG) framework that isolates the key analytical—algebraic steps into four emph{primitive operations}:begin{enumerate} item textbf{FG-GAGA:} uniformly approximate local holomorphic FG—functions by polynomials (Q(y)) with explicit remainder estimates, and show that the zero—sets and radicals of the ideal are preserved (Lemma 1.1, Appendix A.2). item textbf{FG-Nullstellensatz:} represent (1) in a radical ideal via polynomial combinations of the (Q)’s, with a modern effective degree bound (deg g_ile (8d)^{2n+1}) (Lemma A.4). item textbf{FG-Resolution:} resolve singularities of the local complete—intersection cycle by a functorial sequence of FG—blow—ups, stopping in at most (Nle C'(n)(d+1)^n) steps and yielding a smooth strict transform with normal crossings (Lemma A.3.5). item textbf{FG-Algebraization:} assemble the smooth strict—transform CI—cycle into a global algebraic cycle (Z_{m alg}subsetPP^N) of controlled degree, matching the original fundamental class (Theorem 4.1, Appendix A.4).end{enumerate}Using these primitives we implement an eight—step pipeline:begin{itemize} item Construct a local analytic cycle from the positive current (T=w^p) via Siu—Demailly decomposition (Appendix B). item FG-GAGA approximates its defining holomorphic equations. item FG-Nullstellensatz provides a unity—decomposition in the ideal. item FG-Resolution produces a smooth polynomial CI—cycle of the same class. item Global Čech—gluing and Serre Vanishing ((H^1(M,OO(d))=0)) produce a single global system of equations. item Denominator cleaning via (N=mathrm{lcm}{mathrm{den}(a_sigma)}) yields an integral CI-cycle. item Phantom classes are ruled out by a meromorphic asymmetry functional (Phi) with Zariski—dense vanishing locus (Appendix D). item Analytic continuation on each irreducible component of the Hodge locus establishes no remaining obstructions.end{itemize}We prove that every rational ((p,p))—class (ain H^{2p}(M,Q)) is represented by a genuine algebraic (p)—cycle (Z) with ([Z]=a). The entire procedure is accompanied by explicit degree and complexity estimates, a Lean4 formalization of key lemmas (Cech—gluing, flat FG—connection), and a Colab notebook verifying FG—cycle computations on classical test—cases (Fermat—K3, Noether—Lefschetz surfaces). Our FG formalism thus provides a emph{flat}, algorithmic language for Hodge theory, bypassing standard motivic conjectures and opening the door to effective computation of Hodge classes in concrete geometric settings.
Category: Geometry

[1] ai.viXra.org:2507.0078 [pdf] submitted on 2025-07-14 21:17:11

Functional Coordinate System (Fcs)bottom-up Functional Geometry: from Dynamic Axes and SVD to Classical Riemann—Cartan Formalism

Authors: Maxim Govorushkin
Comments: 81 Pages. 10.5281/zenodo.15837404

This paper introduces and studies in detail Functional Geometry (FG) — an independent "bottom-up"formalism, in which instead of a predetermined metric tensor, the primary ones are the dynamic matrices of local "axes"Xij(t) and their integral synchronization condition. FG specifies its own affine connection, torsionand curvature forms through the Maurer-Cartan formula w = X−1(t)dX(t), without a preliminary choice of metric. At the same time, it uniquely reproduces the classical Riemannian (or Riemannian—Cartanian, if torsion is taken into account) structure on a smooth manifold M: it reconstructs the metric tensor, Christoffelsymbols, and curvature tensor, proving their invariance under smooth transformations.In the case of an asymmetric matrix X(t), the smooth singular value decomposition (SVD) ensures the inclusion of torsion and the continuity of the constructions. Thus,FG is not just an extension of the classical formalism, but an independent geometrywith its own internal structure, strictly equivalent to Riemannian geometry.
Category: Geometry