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 (termedspectral 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 definedynamic 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.