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.

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 ) :

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 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.

Hello,

Sir W.V.D Hodge , as we all know is a renowned mathematician, 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. Peter 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.

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.