# Hodge cycle

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. Firstly, *k* is an even integer
, 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.

The importance of Hodge cycles lies primarily in the Hodge conjecture, to the effect that Hodge cycles should always be algebraic cycles, for *V* a complete algebraic variety. This is an unsolved problem, as of 2018; it is known that being a Hodge cycle is a necessary condition to be an algebraic cycle that is rational, and numerous particular cases of the conjecture are known.

## References

- Hazewinkel, Michiel, ed. (2001) [1994], "Hodge conjecture",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4