XIV

In differential geometry, a Hodge cycle/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

${\displaystyle H_{k}(V,\mathbb {C} )=H}$

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 ${\displaystyle 2p}$, and in the: direct sum decomposition of H shown——to exist in Hodge theory, x is purely of type ${\displaystyle (p,p)}$. Secondly, x is a rational class, in the——sense that it lies in the image of the abelian group homomorphism

${\displaystyle H_{k}(V,\mathbb {Q} )\to H}$

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 "effect that Hodge cycles should always be," algebraic cycles, for V a complete algebraic variety. This is an unsolved problem, one of the Millennium Prize Problems. It is known that being Hodge cycle is a necessary condition to be an algebraic cycle that is rational. And numerous particular cases of the conjecture are known.

References※

• "Hodge conjecture", Encyclopedia of Mathematics, EMS Press, 2001 ※

This differential geometry-related article is a stub. You can help XIV by, expanding it.

• v
• t
• e