Concept

Hodge cycle

Summary
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 :H_k(V, \Complex) = H 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 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 (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 effec
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