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. 



Equivalence between statements of Hodge conjecture
The rotation number in question has to do with the behaviour of a differential form $\omega$ on the manifold $X$ under rotation $e^{i\theta}$ of tangent vectors : a complex $k$-form $\omega$ has rotation number $p-q$ iff $$\omega(e^{i\theta}v\_1,\dots,e^{i\theta}v\_k)=e^{i(p-q)\theta}\omega(v\_1,\dots,v\_k)$$ (with $p+q=k$, of course). 


This should explain the terminology, although it is not much in use today. Such a form is called a $(p,q)$-form, their space is denoted $\Omega^{p,q}(X)$, and the subspace of $H^{p+q}(X,\mathbb{C})$ (de Rham cohomology) of classes having a (closed) $(p,q)$-form representative is denoted $H^{p,q}(X)$. 


Hodge theory then gives a decomposition $H^k(X,\mathbb{C})=\bigoplus\_{p+q=k} H^{p,q}(X)$. 


Now a topological cycle $C$ (or its class) of (real) codimension $2p$ has a *Poincaré dual* de Rham cohomology class $[\omega]$, for a closed $2p$ form $\omega$, such that intersection of cycles of *dimension* $2p$ with $C$ coincides with integration of $\omega$.


"The rotation number of $C$ is zero" means that one can choose $\omega$ of type $(p,p)$.


On the other hand, a cohomology class in $H^{2p}(X,\mathbb{Q})$ is said to be Hodge if under the Hodge decomposition
$$H^{2p}(X,\mathbb{Q})\otimes\mathbb{C}\simeq \bigoplus\_{i=-p}^{p} H^{p-i,p+i}(X)$$
it belongs to $H^{p,p}(X)$.


I hope this clarifies the equivalence of the two statements you found.


