Hello,

W.V.D Hodge is famous for his Hodge conjecture. It stands as the open problem in the entry of Millennium prize problems. So for a mathematician to proclaim a conjecture, there must be some heuristic argument or some intuitive explanation. For e.g. Swinnerton-Dyer was looking at the product $\prod Np/p$ on EDSAC and then formulated the rank conjecture ( Now termed as Birch and Swinnerton-dyer conjecture ).

So in a similar way, Hodge might be having some rough heuristics or ideas that led him to the formulation of that conjecture. I am looking for the history and background behind the formulation of Hodge Conjecture. I hope there may be some people in this Community , who have attended the original lectures given by Hodge in ICM. So they can kindly state 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 I want to hear from my learned companions of MO :

1. How are both (Dan's version and Deligne's version ) versions equivalent to each other ? One speaks about the rotation number, where as other version speaks about direct representation of Hodge class.
2. How did Hodge arrive to that conclusion ? Are there any heuristic reasons or intuitive arguments that gives him some hope to conjecture in that direction ? .
3. How can one state analogue of Hodge conjecture in number theory ? Are there any such attempts done previously to formulate what should be the analogue in that case ? .

P.S. : I can believe that conjectures in Number theory can be a result of random attempts and trails, but conjectures to be formulated in Algebraic Geometry and Topology should have a concrete basis, I am looking for that basis . I am very curios to hear the answers for all the three of them , even though they are claimed to be highly techincal in their nature.

Thank you all.