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.

- A fantastic description given by Prof.Dan Freed (here), which an undergraduate student can also understand.
- A bit tough description given by Prof.Pierre Deligne (here), 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*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.