# Hodge conjecture

The statement that for any smooth projective variety $X$ over the field $\mathbf C$ of complex numbers and for any integer $p \geq 0$ the $\mathbf Q$- space $H ^ {2p} ( X, \mathbf Q ) \cap H ^ {p,p}$, where $H ^ {p,p}$ is the component of type $( p, p)$ in the Hodge decomposition

$$H ^ {2p} ( X, \mathbf Q ) \otimes _ {\mathbf Q } \mathbf C = \ \oplus _ {r = 0 } ^ { 2p } H ^ {r, 2p - r } ,$$

is generated by the cohomology classes of algebraic cycles of codimension $p$ over $X$. This conjecture was put forth by W.V.D. Hodge in .

In the case $p = 1$, the Hodge conjecture is equivalent to the Lefschetz theorem on cohomology of type $( 1, 1)$. The Hodge conjecture has also been proved for the following classes of varieties:

1) $X$ is a smooth four-dimensional uniruled variety, that is, a variety such that there exists a rational mapping of finite degree $P ^ {1} \times Y \rightarrow X$, where $Y$ is a smooth variety (see ). Uniruled varieties are, for example, the unirational varieties and the four-dimensional complete intersections with an ample anti-canonical class (see ).

2) $X$ is a smooth Fermat hypersurface of prime order (see , ).

3) $X$ is a simple five-dimensional Abelian variety (see ).

4) $X$ is a simple $d$- dimensional Abelian variety, and $\mathop{\rm End} ( X) \otimes _ {\mathbf Z } \mathbf R = \mathbf R ^ {l}$, where $d/l$ is an odd number, or $\mathop{\rm End} ( X) \otimes _ {\mathbf Z } \mathbf R = [ M _ {2} ( \mathbf R )] ^ {l}$, where $d/2l$ is an odd number.

How to Cite This Entry:
Hodge conjecture. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hodge_conjecture&oldid=47239
This article was adapted from an original article by S.G. Tankeev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article