[W. V. D. Hodge](http://en.wikipedia.org/wiki/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:** 1. How are Freed's version and Deligne's version versions equivalent? 2. 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? . 3. 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.