A Hodge cycle, also called a Hodge class, is a rational cohomology class of type 
on a smooth projective complex algebraic variety 
. In the cohomological convention, the
space of Hodge cycles of codimension 
is
 |
where the intersection is taken in 
and 
is the corresponding summand of the Hodge
decomposition (Deligne).
The cohomology class of every algebraic cycle of codimension 
is a Hodge cycle. The Hodge
conjecture asserts that every Hodge cycle is a rational linear
combination of these cohomology classes. Thus
a Hodge cycle is a class, rather than a specified geometric algebraic
cycle.
