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Hodge cycle

In [mathematics] , a Hodge cycle 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 , C ) = 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 2 p , 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 ) → 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 effect that Hodge cycles should always be [algebraic cycle] s, for V a [complete algebraic variety] . This is an unsolved problem, ; 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.

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