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- Hodge_cycle abstract "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 Kaehler manifold. A homology class x in a homology groupHk(V, C) = Hwhere V is a non-singular complex algebraic variety or Kaehler 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 homomorphismHk(V, Q) → Hdefined 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 2015; 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.".
- Hodge_cycle wikiPageID "898010".
- Hodge_cycle wikiPageLength "1594".
- Hodge_cycle wikiPageOutDegree "17".
- Hodge_cycle wikiPageRevisionID "679169595".
- Hodge_cycle wikiPageWikiLink Algebraic_cycle.
- Hodge_cycle wikiPageWikiLink Algebraic_topology.
- Hodge_cycle wikiPageWikiLink Algebraic_variety.
- Hodge_cycle wikiPageWikiLink Category:Hodge_theory.
- Hodge_cycle wikiPageWikiLink Complete_algebraic_variety.
- Hodge_cycle wikiPageWikiLink Complete_variety.
- Hodge_cycle wikiPageWikiLink Complex_number.
- Hodge_cycle wikiPageWikiLink Differential_geometry.
- Hodge_cycle wikiPageWikiLink Direct_sum_of_modules.
- Hodge_cycle wikiPageWikiLink Hodge_conjecture.
- Hodge_cycle wikiPageWikiLink Hodge_theory.
- Hodge_cycle wikiPageWikiLink Homology_(mathematics).
- Hodge_cycle wikiPageWikiLink Homology_class.
- Hodge_cycle wikiPageWikiLink Homology_group.
- Hodge_cycle wikiPageWikiLink Kaehler_manifold.
- Hodge_cycle wikiPageWikiLink Kähler_manifold.
- Hodge_cycle wikiPageWikiLink Modulo_(jargon).
- Hodge_cycle wikiPageWikiLink Necessary_condition.
- Hodge_cycle wikiPageWikiLink Necessity_and_sufficiency.
- Hodge_cycle wikiPageWikiLink Non-singular.
- Hodge_cycle wikiPageWikiLink Singular_point_of_an_algebraic_variety.
- Hodge_cycle wikiPageWikiLink Universal_coefficient_theorem.
- Hodge_cycle wikiPageWikiLinkText "Hodge cycle".
- Hodge_cycle wikiPageWikiLinkText "absolute Hodge cycles".
- Hodge_cycle hasPhotoCollection Hodge_cycle.
- Hodge_cycle id "h/h047460".
- Hodge_cycle title "Hodge conjecture".
- Hodge_cycle wikiPageUsesTemplate Template:As_of.
- Hodge_cycle wikiPageUsesTemplate Template:Springer.
- Hodge_cycle subject Category:Hodge_theory.
- Hodge_cycle hypernym Kind.
- Hodge_cycle type Article.
- Hodge_cycle type Article.
- Hodge_cycle type Method.
- Hodge_cycle comment "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 Kaehler manifold. A homology class x in a homology groupHk(V, C) = Hwhere V is a non-singular complex algebraic variety or Kaehler 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).".
- Hodge_cycle label "Hodge cycle".
- Hodge_cycle sameAs m.03mx5z.
- Hodge_cycle sameAs Q5876056.
- Hodge_cycle sameAs Q5876056.
- Hodge_cycle wasDerivedFrom Hodge_cycle?oldid=679169595.
- Hodge_cycle isPrimaryTopicOf Hodge_cycle.