Spectral Analysis of Hodge Cycles: A Novel Approach to the Hodge Conjecture via Generalized Moments

Abstract

The Hodge Conjecture, posits a profound connection between the topology and algebraic geometry of complex algebraic varieties. It asserts that Hodge cycles, specific elements in the cohomology of a K\"ahler variety with rational properties, originate from algebraic subvarieties. This paper introduces a novel approach to investigate this conjecture by generalizing the concept of Zernike moments through the lens of harmonic analysis and spectral geometry. Our core idea involves defining a ``characteristic form'' $\eta_Z$ for a Hodge cycle $Z$ within a K\"ahler variety $X$, and expanding this form in terms of the eigenfunctions of the Laplace-Beltrami operator on $Z$. We hypothesize that for algebraic Hodge cycles, the coefficients of this spectral expansion (termed ``spectral fingerprints'') will exhibit specific algebraic patterns, such as being rational numbers, algebraic numbers, or algebraic functions of moduli parameters. We illustrate the computational methodology with a simplified ``toy model'' using a characteristic function on a torus, demonstrating how such coefficients can indeed be rational in a controlled setting. We then outline a conceptual framework for applying this approach to more complex scenarios, specifically K3 surfaces, by leveraging the theory of variations of Hodge structures and moduli spaces to define ``dynamic characteristic forms'' and analyze the algebraic nature of their coefficients. This framework promises to open new avenues in understanding the Hodge Conjecture by translating a deep geometric problem into a question about the algebraic properties of spectral data.