About: Tautological bundle

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In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of -dimensional subspaces of , given a point in the Grassmannian corresponding to a -dimensional vector subspace , the fiber over is the subspace itself. In the case of projective space the tautological bundle is known as the tautological line bundle. Tautological bundles are constructed both in algebraic topology and in algebraic geometry. In algebraic geometry, the tautological line bundle (as invertible sheaf) is

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rdf:type	yago:WikicatVectorBundles yago:Abstraction100002137 yago:Collection107951464 yago:Group100031264 yago:Package108008017
rdfs:label	Tautologisches Bündel (de) 보편 가역층 (ko) Tautological bundle (en)
rdfs:comment	In den mathematischen Gebieten der Topologie und Geometrie ist das tautologische Bündel auf einem projektiven Raum ein Objekt, das jedem Punkt die Gerade zuordnet, aus der er entstanden ist. (de) 대수기하학과 미분기하학에서, 보편 가역층(普遍可逆層, 영어: universal invertible sheaf, tautological invertible sheaf) 또는 보편 선다발(普遍線다발, 영어: universal line bundle, tautological line bundle)은 사영 공간 위에 정의되는 표준적인 가역층(선다발)이며, 보통 로 표기된다. 대략, 사영 공간은 벡터 공간의 원점을 지나는 1차원 부분 벡터 공간들의 모듈라이 공간이므로, 보편 가역층은 사영 공간의 각 점에, 이 점이 나타내는 1차원 부분 벡터 공간을 대응시키는 선다발이다. (ko) In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of -dimensional subspaces of , given a point in the Grassmannian corresponding to a -dimensional vector subspace , the fiber over is the subspace itself. In the case of projective space the tautological bundle is known as the tautological line bundle. Tautological bundles are constructed both in algebraic topology and in algebraic geometry. In algebraic geometry, the tautological line bundle (as invertible sheaf) is (en)
dcterms:subject	Vector bundles
Wikipage page ID	2913502 (xsd:integer)
Wikipage revision ID	1110446329 (xsd:integer)
Link from a Wikipage to another Wikipage	Proj Projective space Projective bundle Borel's theorem Characteristic class Universal bundle Vector space Canonical (disambiguation) Invertible sheaf John Wiley & Sons Thom space Mathematics Classifying space Grassmannian Möbius strip Linear forms Complex projective space Picard group Vector bundles Weil divisor Dual bundle Line bundle Linear subspace Addison-Wesley Algebraic Geometry (book) Algebraic geometry Ample line bundle Dual space Fiber bundle Paracompact space Dimension (vector space) Direct limit Grassmann bundle Linear functional Chern class Blowing up Tautological line bundle Disjoint union Picard group Infinite cyclic Michael Atiyah Vector bundle Euler sequence Exceptional divisor Algebraic vector bundle Stiefel-Whitney class Canonical class Springer-Verlag Hyperplane divisor Serre's twisting sheaf Relative Spec Complex field Homogeneous coordinate Hopf bundle Thom spectrum dbr:Bott_generator
sameAs	Tautological bundle Tautological bundle Tautological bundle Tautological bundle Tautological bundle Tautological bundle
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Citation_needed dbt:Reflist
has abstract	In den mathematischen Gebieten der Topologie und Geometrie ist das tautologische Bündel auf einem projektiven Raum ein Objekt, das jedem Punkt die Gerade zuordnet, aus der er entstanden ist. (de) In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of -dimensional subspaces of , given a point in the Grassmannian corresponding to a -dimensional vector subspace , the fiber over is the subspace itself. In the case of projective space the tautological bundle is known as the tautological line bundle. The tautological bundle is also called the universal bundle since any vector bundle (over a compact space) is a pullback of the tautological bundle; this is to say a Grassmannian is a classifying space for vector bundles. Because of this, the tautological bundle is important in the study of characteristic classes. Tautological bundles are constructed both in algebraic topology and in algebraic geometry. In algebraic geometry, the tautological line bundle (as invertible sheaf) is the dual of the hyperplane bundle or Serre's twisting sheaf . The hyperplane bundle is the line bundle corresponding to the hyperplane (divisor) in . The tautological line bundle and the hyperplane bundle are exactly the two generators of the Picard group of the projective space. In Michael Atiyah's "K-theory", the tautological line bundle over a complex projective space is called the standard line bundle. The sphere bundle of the standard bundle is usually called the Hopf bundle. (cf. .) More generally, there are also tautological bundles on a projective bundle of a vector bundle as well as a Grassmann bundle. The older term canonical bundle has dropped out of favour, on the grounds that canonical is heavily overloaded as it is, in mathematical terminology, and (worse) confusion with the canonical class in algebraic geometry could scarcely be avoided. (en) 대수기하학과 미분기하학에서, 보편 가역층(普遍可逆層, 영어: universal invertible sheaf, tautological invertible sheaf) 또는 보편 선다발(普遍線다발, 영어: universal line bundle, tautological line bundle)은 사영 공간 위에 정의되는 표준적인 가역층(선다발)이며, 보통 로 표기된다. 대략, 사영 공간은 벡터 공간의 원점을 지나는 1차원 부분 벡터 공간들의 모듈라이 공간이므로, 보편 가역층은 사영 공간의 각 점에, 이 점이 나타내는 1차원 부분 벡터 공간을 대응시키는 선다발이다. (ko)
prov:wasDerivedFrom	wikipedia-en:Tautological_bundle?oldid=1110446329&ns=0
page length (characters) of wiki page	14273 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Tautological_bundle
is Link from a Wikipage to another Wikipage of	Algebraic geometry of projective spaces Universal bundle Stiefel–Whitney class Glossary of algebraic geometry Grassmannian Fubini–Study metric Jumping line Algebraic variety Topological K-theory Cobordism Tautological Tautological line bundle Tautological one-form Moduli space Real projective space

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