[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 ) :** 



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



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> 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. 
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**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.



Heuristics for the Hodge Conjecture
The best answer I can imagine for a question like this is to quote the man himself:
"The second result of Lefschetz tells us that a necessary and sufficient condition
that a 2-cycle $\Gamma\_2$ in $V\_2$ be algebraic... This result has many geometrical applications...
It is clearly a matter of great importance to extend Lefschetz's condition for a
2-cycle to be algebraic. The general problem is as follows...."


See page 184 of
the [Proceedings of the ICM 1950](https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1950.1/ICM1950.1.ocr.pdf)
for the full statement:


*Hodge, W. V. D.*, The topological invariants of algebraic varieties, Proc. Intern. Congr. Math. (Cambridge, Mass., Aug. 30-Sept. 6, 1950) 1, 182-192 (1952). [ZBL0048.41701](https://zbmath.org/?q=an:0048.41701).


