W. V. D. Hodge is famous for his Hodge conjecture, one of the Millennium prize problems. Hodge might have had some rough heuristics or ideas that led him to the formulation of the conjecture.

I am looking for the history and background behind the formulation of Hodge Conjecture. How did Hodge arrive at his conjecture?

**Hodge Conjecture ( What I understood after reading Dan Freed's article ) :**

On a complex projective manifold $\mathbb{X}$, a topological cycle $\mathfrak{C}$ ( on $\mathbb{X}$ ) is homologous to a rational combination of algebraic cycles iff $\mathfrak{C}$ has rotation number $0$. ( Rotation number $e^{i\theta}$ on differential form, volumes are invariant under rotation).

**Hodge Conjecture (Deligne's description):**

On a projective non-singular algebraic variety over $\mathbb{C}$, any Hodge class is a rational combination of classes $\rm{cl(\mathbb{Z})}$ of algebraic cycles.

**The things that interest me:**

- How are Freed's version and Deligne's version versions equivalent?
- How did Hodge arrive at that conclusion? Were there heuristic reasons or intuitive arguments that gives him some hope for a conjecture in that direction? .
- How can one state an analogue of the Hodge conjecture in number theory? Are there any attempts to formulate an analogue in that case?

I am curious to hear answers, even if highly technical in nature.

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