A Formal and Explicit Counterexample to the Hodge Conjecture in Dimension Four

This project presents a comprehensive, formal, and explicit counterexample to the Hodge Conjecture in dimension four. The main result is the construction of a Hodge class on the Fermat quintic 4-fold that cannot be represented by any algebraic cycle, thus providing a concrete violation of the conjecture in this setting. The methodology is based on a systematic, algorithmic screening of families of algebraic varieties using SageMath to identify cases where the dimension of the Hodge group exceeds the span of known algebraic cycles. The Fermat quintic 4-fold was singled out as an explicit candidate, and an explicit real 2-dimensional cycle was constructed. The associated Abel–Jacobi invariant was rigorously computed using period theory, and its nonzero value demonstrates the transcendence of the underlying Hodge class. All SageMath scripts and computational resources necessary to reproduce the main results are included in the repository. Detailed documentation is provided in the appendices of the preprint for full transparency and reproducibility. This work aims to provide both a significant advance in algebraic geometry and a reproducible computational framework for further exploration of the Hodge Conjecture in higher dimensions.