In differential geometry, a Hodge cycle or Hodge class is a particular kind of homology class defined on a complex algebraic variety V, or more generally on a Kähler manifold.
A homology class x in a homology group where V is a non-singular complex algebraic variety or Kähler manifold is a Hodge cycle, provided it satisfies two conditions.
, and in the direct sum decomposition of H shown to exist in Hodge theory, x is purely of type
Secondly, x is a rational class, in the sense that it lies in the image of the abelian group homomorphism defined in algebraic topology (as a special case of the universal coefficient theorem).
The conventional term Hodge cycle therefore is slightly inaccurate, in that x is considered as a class (modulo boundaries); but this is normal usage.