About: Lefschetz hyperplane theorem

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In mathematics, specifically in algebraic geometry and algebraic topology, the Lefschetz hyperplane theorem is a precise statement of certain relations between the shape of an algebraic variety and the shape of its subvarieties. More precisely, the theorem says that for a variety X embedded in projective space and a hyperplane section Y, the homology, cohomology, and homotopy groups of X determine those of Y. A result of this kind was first stated by Solomon Lefschetz for homology groups of complex algebraic varieties. Similar results have since been found for homotopy groups, in positive characteristic, and in other homology and cohomology theories.

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rdf:type	Thing yago:WikicatTheoremsInAlgebraicGeometry yago:WikicatTheoremsInAlgebraicTopology yago:WikicatTopologicalMethodsOfAlgebraicGeometry yago:Ability105616246 yago:Abstraction100002137 yago:Cognition100023271 yago:Communication100033020 yago:Know-how105616786 yago:Message106598915 yago:Method105660268 yago:Proposition106750804 yago:PsychologicalFeature100023100 yago:Statement106722453 yago:Theorem106752293
rdfs:label	Satz von Lefschetz über Hyperebenenschnitte (de) Théorème des hyperplans de Lefschetz (fr) Lefschetz hyperplane theorem (en) 렙셰츠 초평면 정리 (ko) レフシェッツ超平面定理 (ja) Lefschetzs hyperplansats (sv)
rdfs:comment	In der Mathematik, speziell in der Algebraischen Geometrie und Algebraischen Topologie, stellt der Satz von Lefschetz über Hyperebenenschnitte einen Zusammenhang zwischen der Gestalt einer algebraischen Varietät und der Gestalt ihrer Untervarietäten her. Er besagt, dass für einen Hyperebenenschnitt in einer projektiven Varietät die Homotopie-, Homologie- und Kohomologiegruppen bis zu einer gewissen Dimension bereits durch diejenigen von festgelegt sind. Benannt ist die Aussage nach Solomon Lefschetz. (de) 대수기하학에서 렙셰츠 초평면 정리(Лефшец超平面定理, 영어: Lefshetz hyperplane theorem)는 복소수 사영 대수다양체의 위상수학과 그 초평면 단면의 위상수학 사이의 관계에 대한 정리이다. (ko) 数学では、特に代数幾何学や代数トポロジーでは、レフシェッツの超平面定理(Lefschetz hyperplane theorem)は、代数多様体の形と部分多様体の形の間のある関係についてのステートメントであり、この定理は、射影空間に埋め込まれた多様体 X と(hyperplane section) Y に対し、X のホモロジー、コホモロジー、ホモトピー群は、Y のそれらをも決定するという定理である。この種類の結果は、最初に複素代数多様体のホモロジー群に対し、ソロモン・レフシェッツ(Solomon Lefschetz)により言明された。同様の結果が、正の標数でも、他のホモロジー、コホモロジー理論で、ホモトピー群に対して発見されている。なお、レフシェッツ超平面定理のことを弱レフシェッツ定理(Weak Lefschetz Theorem)とも言う。 (ja) Lefschetzs hyperplansats är inom matematiken ett precist uttalande om vissa relationer mellan formen av en algebraisk varietet och formen av dess delvarieteter. Mer precist säger satsen att om X är en varietet inbäddad i och en Y, bestämmer homologin, kohomologin och av X motsvarande av Y. Ett resultat av detta slag framlades först av för homologigrupper av komplexa algebraiska varieteter. Likadana resultat har senare upptäckts för homotopigrupper i positiv karakteristik, och i andra homologi- och kohomologiteorier. (sv) In mathematics, specifically in algebraic geometry and algebraic topology, the Lefschetz hyperplane theorem is a precise statement of certain relations between the shape of an algebraic variety and the shape of its subvarieties. More precisely, the theorem says that for a variety X embedded in projective space and a hyperplane section Y, the homology, cohomology, and homotopy groups of X determine those of Y. A result of this kind was first stated by Solomon Lefschetz for homology groups of complex algebraic varieties. Similar results have since been found for homotopy groups, in positive characteristic, and in other homology and cohomology theories. (en) En mathématiques, et plus précisément en géométrie algébrique et en topologie algébrique, le théorème des hyperplans de Lefschetz est un énoncé précis de certaines relations entre la forme d'une variété algébrique et la forme de ses sous-variétés. Plus précisément, le théorème énonce que pour une variété X plongée dans l'espace projectif et une section hyperplane (i.e. une intersection de X à un hyperplan) Y, les groupes d'homologie, de cohomologie et d'homotopie de X déterminent ceux de Y. Un résultat de ce type a été énoncé pour la première fois par Solomon Lefschetz pour les groupes d'homologie de variétés algébriques complexes. Des résultats similaires ont depuis été trouvés pour les groupes d'homotopie, en caractéristique positive et dans d'autres théories d'homologie et de cohomologi (fr)
rdfs:seeAlso	Lefschetz manifold
dct:subject	Topological methods of algebraic geometry Theorems in algebraic topology Theorems in algebraic geometry Morse theory
Wikipage page ID	3459657 (xsd:integer)
Wikipage revision ID	1117693748 (xsd:integer)
Link from a Wikipage to another Wikipage	Cambridge University Press Princeton University Press Projective space Decomposition theorem Algebraic topology Andreotti–Frankel theorem Annals of Mathematics René Thom Universal coefficient theorem Intersection homology Mathematics Morse theory Constructible sheaf Topological methods of algebraic geometry Long exact sequence Étale cohomology Hopf surface Picard group Michigan Mathematical Journal Publications Mathématiques de l'IHÉS Theorems in algebraic topology Affine variety Aldo Andreotti Alexander Grothendieck Algebraic geometry Algebraic variety Lefschetz pencil Theodore Frankel Hyperplane section Theorems in algebraic geometry Cohomology Cohomology ring Homology (mathematics) Homotopy group Donald C. Spencer Solomon Lefschetz Morse theory Kunihiko Kodaira Kähler manifold Michael Artin Raoul Bott Relative homology Morse lemma Springer-Verlag Akizuki–Nakano vanishing theorem Cohomology class CW-complex
Link from a Wikipage to an external page	http://www.numdam.org/item%3Fid=PMIHES_1980__52__137_0 https://web.archive.org/web/20040707020702/http:/www.math.polytechnique.fr/cmat/sabbah/hodge-str.pdf http://www.math.polytechnique.fr/cmat/sabbah/livres/hodge-str.pdf http://projecteuclid.org/DPubS/Repository/1.0/Disseminate%3Fview=body&id=pdf_1&handle=euclid.mmj/1028998225
sameAs	Lefschetz hyperplane theorem Lefschetz hyperplane theorem Lefschetz hyperplane theorem Lefschetz hyperplane theorem Lefschetz hyperplane theorem Lefschetz hyperplane theorem Lefschetz hyperplane theorem Lefschetz hyperplane theorem Lefschetz hyperplane theorem
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Nowrap_begin dbt:Nowrap_end dbt:See_also dbt:Harvs
authorlink	Pierre Deligne (en)

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