Dear everyone, 

I was unable to obtain the equivalence between the two statements of the Hodge conjecture. I searched for some previous questions that others asked here, to check whether someone has asked the same thing previously. But I didn't find any such instance, that is why I am asking. 
We know that Hodge conjecture gives some relation between the topological cycles and algebraic cycles. But I have read two different variations of the same conjecuture. I number my pointers. 

 1. A fantastic description given by Prof.Dan Freed ([here](http://www.ma.utexas.edu/users/dafr/HodgeConjecture/netscape_noframes.html)), which an undergraduate student can also understand. 
 2.  A bit tough description given by Prof.Pierre Deligne ([here](http://www.claymath.org/millennium/Hodge_Conjecture/Official_Problem_Description.pdf)), with lot of technical terms and constructions. 

So I was befuddled in asking myself that how can one obtain equivalence between the both statements. 

**Dan Freed's Version :** 

> He considers a Topological cycle ( boundary less chains that are free to deform ) on a projective manifold. Then he says that the topological cycle is homologous to a rational combination of algebraic cycles, if and only if the topological cycle has a rotation number Zero. 

**P.Deligne's Version :** 

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


So now I have the following queries for my learned friends.

 - How can one explain that both the statements are equivalent to each other ?  One speaks about the rotation number and another doesn't even speak about it. How can one say that both the statements are valid ?  I infact know that both the statements are valid ( as both the speakers are seminal mathematicians ) But how ? 
 - So can anyone explain me what the *[Rotation number](http://en.wikipedia.org/wiki/Rotation_number)* has to do with the Hodge Conjecture ? I obtained some information about the rotation number from Wiki. But I am afraid , to decide whether Freed is speaking about the same rotation number ( given in wiki ) in his talk ? or something different ? 

I would be really honored to hear answers for both of them . Thank you one and all for sparing your time reading my question.