Hodge cycle

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 cycles, for "V" a complete algebraic variety. This is an unsolved problem, as of 2004; 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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