The Hodge Conjecture

Importance: Outstanding ✭✭✭✭
Author(s): Hodge, W. V. D.
Recomm. for undergrads: no
Prize: $1,000,000 from the Clay Mathematics Institute
Posted by: Charles
on: July 13th, 2008

\begin{conjecture} Let $X$ be a complex projective variety. Then every Hodge class is a rational linear combination of the cohomology classes of complex subvarieties of $X$. \end{conjecture}

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A complex projective variety is the set of zeros of a finite collection of homogeneous polynomials on projective space, and we are concerned with the singular cohomology ring. There is a well known Hodge Decomposition of the cohomology into groups $H^{p,q}(X.\mathbb{C})$ which hare holomorphic in $p$ variables and antiholomorphic in $q$ variables with the property that $\oplus_{p+q=k}H^{p,q}=H^k$.

So we define the Hodge classes to be those in the intersection $H^{k,k}(X,\mathbb{C})\cap H^{2k}(X,\mathbb{Q})$. It is fairly easy to show that the cohomology class of a subvariety is Hodge. We say that a cycle is \emph{algebraic} if it is a rational linear combination of the classes of subvarieties. So every algebraic cycle is Hodge. In dimension one, we have the following result:

\begin{theorem}[Lefshetz (1,1) Theorem] Any element of $H^2(X,\mathbb{Q})\cap H^{1,1}$ is the cohomology class of a divisor, and so is algebraic. \end{theorem}

It's also true that if the Hodge Conjecture holds for cycles of degree $p2n-p$. So this and the (1,1) Theorem show that the Hodge Conjecture is true for complex curves, surfaces and threefolds.


% Example: %*[B] Claude Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10 (1961), 114. % %[CRS] Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas: \arxiv[The strong perfect graph theorem]{math.CO/0212070}, % Ann. of Math. (2) 164 (2006), no. 1, 51--229. \MRhref{MR2233847} % % (Put an empty line between individual entries)

*[Hod] Hodge, W. V. D. "The topological invariants of algebraic varieties". Proceedings of the International Congress of Mathematicians, Cambridge, MA, 1950, vol. 1, pp. 181–192.

* indicates original appearance(s) of problem.