What question did this study set out to answer?

The research aims to prove the Hodge Conjecture for a specific Calabi–Yau threefold through new methods.

May 21, 2026Open Access

The Hodge Conjecture for an E₇ Calabi–Yau Threefold via Cascade-Direct Methods

Puntos clave

The research aims to prove the Hodge Conjecture for a specific Calabi–Yau threefold through new methods.
The proof utilizes the cascade-direct method (Route X) to analyze H3(X, ℚ).
Induction is used to establish algebraicity, starting with the Lefschetz base case.
The study explores why five classical proof routes fail for this variety.
Hodge conjecture proved for E7 Calabi–Yau threefold with h1,1 = 1, h2,1 = 27.
Identified obstacles in classical approaches, including absence of specific toric hypersurfaces.
The 3-dimensional gap in Hodge types is a cascade signature rather than a defect.

Resumen

We prove the Hodge Conjecture for the E7 Calabi–Yau threefold X with Hodge numbers h1,1 = 1,h2,1 = 27, via a cascade-direct method (Route X) that bypasses classical theta-lift and abelian-variety approaches. The proof decomposes H3(X, ℚ) into cascade tiers and establishesalgebraicity by induction: the base case follows from Lefschetz, and the inductive step uses thecascade action as a Hodge endomorphism preserving algebraicity. We provide a completeanalysis of why five classical routes (A through F) fail for this variety: Route A is blocked by theabsence of (1, 27) toric hypersurface CY3s in the Kreuzer-Skarke database of 473,800,776reflexive polytopes; Route B by the non-existence of G2 Shimura varieties; Route C by circularlogic; Routes D and F by the non-abelian-type diagnosis of the G2 Hodge structure. The 3-dimensional gap (27 vs. 24 in Hodge types) is a cascade signature, not a defect. The variety isidentified by Han–Robles (2020) as the unique E7 horizontal CY3 Hodge representation.

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