Summary of formalized insights. Implications for resolving HC and broader conjectural mathematics.
References
Comprehensive list, including classics (Deligne, Voisin) and recent works (e.g., spectral analysis, formalization in Lean).
I. Introduction
A. Background on the Hodge Conjecture and Millennium Problems
In the year 2000, the Clay Mathematics Institute established the Millennium Prize Problems, a collection of seven profound unsolved questions in mathematics, each carrying a prize of one million dollars for a correct resolution. These problems were selected for their depth, breadth, and potential to catalyze transformative advances across mathematical disciplines. They encompass diverse areas: the Navier--Stokes equations in fluid dynamics, the P versus NP problem in computational complexity, the Riemann Hypothesis in number theory, the Birch and Swinnerton-Dyer Conjecture in arithmetic geometry, the Yang--Mills existence and mass gap in quantum field theory, the Poincar Conjecture (resolved by Grigori Perelman in 2003), and the Hodge Conjecture (HC) in algebraic geometry.
The Hodge Conjecture, formulated by William Vallance Douglas Hodge in the mid-20th century, stands as a cornerstone of algebraic geometry, bridging topology, algebra, and geometry. Let \(X\) be a smooth projective variety over \(\mathbb{C}\) of complex dimension \(n\). The singular cohomology group \(H^{2p}(X, \mathbb{Q})\) admits a Hodge decomposition over \(\mathbb{C}\):
\[
H^{2p}(X, \mathbb{C}) = \bigoplus_{r+s=2p} H^{r,s}(X),
\]
where \(H^{r,s}(X)\) comprises classes representable by differential forms of type \((r,s)\). A rational class \(\gamma \in H^{2p}(X, \mathbb{Q})\) is a Hodge class if its complexification lies in \(H^{p,p}(X)\). Algebraic cycles of codimension \(p\), forming the group \(Z^p(X)\), induce classes via the cycle class map:
