![](/files/happy5.png)
The Hodge Conjecture
![$ X $](/files/tex/302cdeba125e821f3406302c9789229d48f42ea7.png)
![$ X $](/files/tex/302cdeba125e821f3406302c9789229d48f42ea7.png)
A complex projective variety is the set of zeros of a finite collection of homogeneous polynomials on projective space, and we are concerned with the singular cohomology ring. There is a well known Hodge Decomposition of the cohomology into groups which hare holomorphic in
variables and antiholomorphic in
variables with the property that
.
So we define the Hodge classes to be those in the intersection . It is fairly easy to show that the cohomology class of a subvariety is Hodge. We say that a cycle is algebraic if it is a rational linear combination of the classes of subvarieties. So every algebraic cycle is Hodge. In dimension one, we have the following result:
![$ H^2(X,\mathbb{Q})\cap H^{1,1} $](/files/tex/d570fbb275e0d59af23bd33e8c00a62cf48b5ee9.png)
It's also true that if the Hodge Conjecture holds for cycles of degree , then it holds for cycles of degree
. So this and the (1,1) Theorem show that the Hodge Conjecture is true for complex curves, surfaces and threefolds.
Bibliography
*[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.