# The Hodge Conjecture

 Importance: Outstanding ✭✭✭✭
 Author(s): Hodge, W. V. D.
 Subject: Geometry » Algebraic Geometry
 Keywords: Hodge Theory Millenium Problems
 Prize: \$1,000,000 from the Clay Mathematics Institute
 Posted by: Charles on: July 13th, 2008
Conjecture   Let be a complex projective variety. Then every Hodge class is a rational linear combination of the cohomology classes of complex subvarieties of .

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:

Theorem  (Lefshetz (1,1) Theorem)   Any element of is the cohomology class of a divisor, and so is algebraic.

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.