Hodge cycle

{{Short description|Kind of homology class in differential geometry}} 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

:<math>H_k(V, \Complex) = H</math>

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 <math>2p</math>, and in the direct sum decomposition of ''H'' shown to exist in Hodge theory, ''x'' is purely of type <math>(p,p)</math>. Secondly, ''x'' is a rational class, in the sense that it lies in the image of the abelian group homomorphism

:<math>H_k(V, \Q) \to H</math>

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

==References== *{{Springer|id=h/h047460|title=Hodge conjecture}}

Category:Hodge theory