# Heuristics for the Hodge Conjecture

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.

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Do you happen to know "Peter" on a first-name basis? Also, are you planning to ask for heuristics behind many more conjectures? – S. Carnahan Jul 2 '12 at 10:45
I've edited the post – Charles Matthews Jul 2 '12 at 11:28
It's worth looking at Hodge's original ICM article. The "conjecture" was announced with little fanfare... (2) He certainly knew of Lefschetz's $(1,1)$ theorem, so I suppose that this was a natural extrapolation. (3) Some people consider Tate's conjecture that the Galois invariant part of $\ell$-adic cohomology is spanned by algebraic cycles to be an analogue. If you search for "Hodge conjecture" here, you'll find a number of useful discussions. – Donu Arapura Jul 2 '12 at 12:39
@Donu- I believe in the Proceedings of a Conference on Arithmetical Algebraic Geometry held at Purdue Tate says that he doesn't think the two conjectures are particularly connected. I can look up exact reference if you wish. – aginensky Jul 3 '12 at 0:31

The best answer I can imagine for a question like this is to quote the man himself: "The second result of Lefschetz tells us that a necessary and sufficient condition that a 2-cycle $\Gamma_2$ in $V_2$ be algebraic... This result has many geometrical applications... It is clearly a matter of great importance to extend Lefschetz's condition for a 2-cycle to be algebraic. The general problem is as follows...."

See page 184 of http://www.mathunion.org/ICM/ICM1950.1/Main/icm1950.1.0182.0192.ocr.pdf for the full statement.

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Thank you donu arapura +1. – Shanmukha_Srinivasan Jul 2 '12 at 15:42

Edited: One point is that Hodge's original version of the conjecture was wrong, and in a couple of ways. You do need rational coefficients (integral is too much to ask for, see ref below). Also a more general conjecture of Hodge fails: see

http://people.math.jussieu.fr/~leila/grothendieckcircle/HodgeConj.pdf

for one of all the all-time great disrespectful titles. (Hodge was a major innovator in algebraic geometry and complex manifold theory: but his theory needed plenty of work, particularly I think from Kodaira, before it became clear foundationally. The same is true, perhaps in spades, for Lefschetz.)

Edit: At the level of linear algebra, Hodge theory deals with a vector space with a double grading. Using the heuristic grading = concept of homogeneity, the subspaces of (p,p) type in Hodge theory can be picked out by the use of the circle group, as Freed does. Really (pun intended) there is a bigger group with two dimensions involved, the group of non-zero complex numbers under multiplication. A shallow remark is that these two groups control the linear algebra in Hodge theory. A deeper remark is that the vector spaces involved also have a rational structure, coming from the concept of integral cycle in topology. The interaction of the groups with the rational structure isn't shallow at all.

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The error corrected in Grothendieck's paper was not the issue of integer versus rational coefficients -- this was corrected in Atiyah and Hirzebruch, "Algebraic Cycles on Complex Manifolds", Topology 1 (1962) p. 25-45 . Grothendieck's correction concerned a conjectural characterization of those classes in $H_k(X)$ which are pushed forward from some algebraic $Y \subset X$ with $\dim Y=m$. (The case h=2m$being the Hodge conjecture.) – David Speyer Jul 2 '12 at 11:57 Ah, thanks, helpful. – Charles Matthews Jul 2 '12 at 12:28 Thank you David, Charles. The error was made clear now. +1, both to your comments and answer – Shanmukha_Srinivasan Jul 2 '12 at 15:43 Of course, I meant$k=2m$, not$h=2m\$, in my comment. – David Speyer Jul 2 '12 at 16:47
And the title is "Analytic cycles..." not "Algebraic cycles...". MR0145560 – David Speyer Jul 2 '12 at 16:50