In differential geometry, a Hodge cycle or Hodge class is a particular kind of

homology class Homology may refer to: Sciences Biology *Homology (biology), any characteristic of biological organisms that is derived from a common ancestor *Sequence homology, biological homology between DNA, RNA, or protein sequences *Homologous chromo ...

defined on a

complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...

algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...

''V'', or more generally on a

Kähler manifold In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arn ...

. A homology class ''x'' in a

homology group In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...

H_k(V, \Complex) = H

where ''V'' is a

non-singular In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In ca ...

complex algebraic variety or Kähler manifold is a Hodge cycle, provided it satisfies two conditions. Firstly, ''k'' is an even integer

2p

, and in the direct sum decomposition of ''H'' shown to exist in Hodge theory, ''x'' is purely of type

(p,p)

. Secondly, ''x'' is a rational class, in the sense that it lies in the image of the abelian group homomorphism :

H_k(V, \Q) \to H

defined in

algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...

(as a special case of the

universal coefficient theorem In algebraic topology, universal coefficient theorems establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological space , its ''integral homology groups'': : completely ...

). 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 In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties. In simple terms, the Hodge conjectu ...

, to the effect that Hodge cycles should always be

algebraic cycle In mathematics, an algebraic cycle on an algebraic variety ''V'' is a formal linear combination of subvarieties of ''V''. These are the part of the algebraic topology of ''V'' that is directly accessible by algebraic methods. Understanding the a ...

s, for ''V'' a complete algebraic variety. This is an unsolved problem, ; it is known that being a Hodge cycle is a

necessary condition In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...

to be an algebraic cycle that is rational, and numerous particular cases of the conjecture are known.

References

* Hodge theory {{Differential-geometry-stub