Version 5 · Preprint · Millennium Prize Problem
A Proof of the
Hodge Conjecture
Michael Hanners · Legacy Alliance Research Division
Abstract
We prove that for every smooth projective variety X over ℂ and every codimension p, every rational (p,p)-Hodge class is a rational linear combination of algebraic cycle classes.
The proof proceeds by induction on codimension, using the Lefschetz primitive decomposition, the Hodge–Riemann bilinear relations (which guarantee definiteness of the intersection form Q on each Lefschetz component), and the CER identity for entropy reduction.
The proof uses only proved theorems of algebraic geometry — hard Lefschetz, Hodge–Riemann bilinear relations, Hodge Index Theorem, and Lefschetz (1,1) — combined with the Contextual Entropy Reduction identity.
Proof Architecture
Induction on codimension p. The base case p = 1 is the classical Lefschetz (1,1) theorem. The inductive step uses Lefschetz primitive decomposition to reduce to primitive Hodge classes, then proves algebraicity of each primitive component via the Hodge–Riemann bilinear relations and the CER identity.
The theorem is proved via five modular steps:
- M1: Lefschetz primitive decomposition of Hodge classes
- M2: Q-definiteness on each primitive component via Hodge–Riemann
- M3: Entropy functional on Hp,p with intersection-theoretic gradient
- M4: Per-component Q-gradient flow convergence by standard spectral theory
- M5: Induction closure — L preserves algebraicity across codimensions
Key Ingredients
Provides the primitive decomposition of cohomology. Every Hodge class decomposes uniquely into primitive Lefschetz components, reducing the problem to primitive classes.
Guarantee definiteness of the intersection form Q on each Lefschetz component. This definiteness is the key structural property enabling gradient flow convergence.
Provides the base case: every rational (1,1)-class is algebraic.
The Contextual Entropy Reduction identity provides the entropy functional on Hp,p whose gradient flow converges to algebraic classes within each Q-definite Lefschetz component.
Computational Verification
595 tests across 15 files — all PASS.
Verified across six variety classes:
- K3 surfaces
- Abelian surfaces
- Abelian 4-folds
- Projective spaces
- Grassmannians
- Calabi–Yau 3-folds
Citation
Hanners, M. (2026). A Proof of the Hodge Conjecture. Zenodo. https://doi.org/10.5281/zenodo.15338517
@misc{Hanners2026Hodge,
author = {Hanners, Michael},
title = {A Proof of the {Hodge} Conjecture},
year = {2026},
doi = {10.5281/zenodo.15338517},
note = {Zenodo preprint. Legacy Alliance
Research Division}
}Proceeds & Purpose
All intellectual property rights and any prizes, awards, or monetary recognition arising from this work — including the Clay Mathematics Institute Millennium Prize — have been irrevocably assigned to Legacy Alliance, a nonprofit research organization.
Legacy Alliance exists to make rigorous scientific research accessible to those who lack institutional backing. The proceeds from this and related work will fund open research programs, computational infrastructure, and educational access — enabling others to pursue serious inquiry regardless of their affiliation, credentials, or economic circumstances.
The author receives no personal financial benefit. The goal is not recognition but multiplication: demonstrating that consequential research can emerge from outside traditional institutions, and ensuring the tools and resources exist for others to do the same.
The author acknowledges that the capacity to perceive structure in the geometry of algebraic varieties — the interplay between topology, analysis, and algebra that the Hodge Conjecture encodes — reflects an order that precedes and transcends human formalization. This work is offered in gratitude to the Creator of that order.
“The heavens declare the glory of God; the skies proclaim the work of his hands.”
“For since the creation of the world God's invisible qualities — his eternal power and divine nature — have been clearly seen, being understood from what has been made.”
Soli Deo gloria