Universal Kernel Operators and Seed Correspondences in the Direction of the Hodge Conjecture
Internal ID: CGI-RSR-000007
Document Type: Research Paper
Publication Date: May 2026
Status: Public
Domains: Mathematics
Research Topics: Algebraic geometry, topology, Hodge theory, Clay Millennium Problems, Hodge Conjecture
Abstract
We study a natural operator associated with the wedge product on exterior powers and show that its kernel admits a universal representation-theoretic structure. In the symplectic setting, the invariant sector reduces to a canonical two-dimensional plane, allowing an explicit description of the kernel and yielding a universal hyperplane relation. We then realize this operator structure geometrically via algebraic correspondences on polarized abelian varieties. This produces a distinguished seed class whose behavior is governed by the operator kernel. On products X^N, these seed classes generate a canonical subspace W_N ⊂ H^{2,2}(X^{N}), which is identified as the irreducible S_N–representation of type (N − 2, 2). We show that the seed generates a correspondence algebra whose action on primitive cohomology produces a graded orbit isomorphic to the symmetric algebra Sym•(W_N). This orbit captures all balanced representations (N −k, k) and provides a representation-theoretic description of the primitive interaction sector. The resulting coherence cycle mechanism identifies the primitive cohomology generated by the seed orbit and yields an explicit algebraic model for its structure. This establishes a direct connection between operator-theoretic kernel structure, algebraic correspondences, and the representation theory of cohomology.
Available Document
Citation:
Petersen, B. L. (2026). Universal Kernel Operators and Seed Correspondences in the Direction of the Hodge Conjecture. Zenodo. https://doi.org/10.5281/zenodo.19970899
Source Code and Supporting Materials
[Supporting documents listed here]
Summary and Notes
The route pursued here was motivated by an operator principle that had appeared productive in earlier work on Birch–Swinnerton–Dyer: difficult invariants from different mathematical descriptions sometimes become comparable when expressed through a common quadratic operator and its kernel structure.
The same heuristic was tested here in a Hodge-theoretic setting. The expectation was that additional specialized machinery might eventually be required. Instead, the operator framework itself continued to organize progressively richer layers of the problem, linking kernel relations, algebraic correspondences, and primitive cohomology generation.
In that sense, Coherence Geometry served primarily as a source of method: it suggested looking for hidden organizing operators rather than treating each cohomological sector independently.
Related Work
Foundations of Coherence Geometry Textbook, Algebraic formulations of CG