The only two vector bundles with base space a circle and onedimensional. A bundle over a manifold is trivial if it is simply the cartesian product of. E 2 of trivialized vector bundles satisfying the condition. In the case there is no ambiguity about the base, we will denote a vector bundle by efor short.
Vector bundles over normal varieties trivialized by finite morphisms. H is an isomorphism of vector bundles satisfying that 4. This new vector bundle fe is called the \associated vector bundle to e, given by the functor f. For any smooth manifold m, e m rk is a trivial bundle over m. If x is a smooth, connected and projective variety over an algebraically closed. An algebraic vector bundle of rank r on x is a linear. Preliminaries if v is a nitedimensional vector space over a eld f, say with dimension n 0, the 1dimensional top exterior power nv understood to mean f if n 0 is sometimes called the determinant of v, and is denoted detv.
The interest of the elliptic curve lies in the fact that it provides the first nontrivial case, grothendieck 6 having shown that for a rational curve every vector bundle is a direct sum of line. X be a vector bundle of rank r over an algebraic variety x. Given a space x, we take a real or complex finite dimensional vector space v and. U ck that is a biholomorphic map of complex manifolds. X y that trivializes v then v is essentially finite. Sections section the projection e m sends an entire r k to a point. A connection on any vector bundle gives a way of parallel transporting sections along curves.
Subbundles and quotient bundles motivation example n. Induced bundles and vector bundles equivalence classes of line bundles and cohomology classes in h1x. It is a rst example of a characteristic class of a ber bundle. A vector bundle together with an orientation is called an oriented bundle. A vector bundle for which a trivialization exists is called trivializable accordingly, the local triviality condition in the definition of topological vector bundles def. The notion of vector bundle is a basic extension to the geometric domain of the fundamental idea of a vector space. Motivation let x,o be a cp premanifold with corners, 0.
Let y be a normal and projective variety over an algebraically closed field k and v a vector bundle over y. Sorry, we are unable to provide the full text but you may find it at the following locations. A vector bundle of rank 1 is usually called a line bundle. The latter is, of course, equivalent to a trivialisation of the vector bundle by definition, if you set things up correctly but the former is most definitely not. The basic invariant of an oriented bundle is the euler class. In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space x for example x could be a topological space, a manifold, or an algebraic variety. As originally phrased, your answer read as if you just needed a local section of the vector bundle.
We have developed the notion of a cp vector bundle over xas a certain kind of cp mapping e xthat is roughly speaking a cp varying family of. Strictly speaking, a vector bundle is an entire triple e. We prove that if there exist a kscheme x and a finite surjective morphism g. A trivial bundle can be trivialized over its entire base space. Suppose that qis nondegenerate on the generic bre of e. Degrees of vector bundles are always taken with respect to the polarization l. Vector bundles of rank 1 is also called the line bundle. The vector bundle of rank rover xis said to be trivial if there exists a global bundle chart e x. The multiplication that is, cup product by the euler class of an oriented bundle gives rise to a gysin sequence.
L with values in some line bundle l over b which is not identically zero on any bre. Induced bundles and vector bundles induced bundles. A kvector bundle over mof rank kconsists of a bundle e m whose. This package contains 57 vector elements, in a wide range of file formats ai8, ai, eps8, eps and png, all saved at a resolution of 300px. Using this result and the theory of chern classes of vector bundles, we give a general criterion for a uniform vector bundle on a homogenous space to be splitting. Accordingly, the local triviality condition in the definition of topological vector bundles def. E is obtained from the cartesian product g x e by identifying gu x e with g x u.
A vector bundle for which a trivialization exists is called trivializable. For x x a suitable space then a vector bundle over x x is called trivial if there is no twist in how the fibers vary over it. The vintage vector bundle is a great pack of handmade resources. Construct a ranktwo vector bundle over the smooth quadric x.
And if you have a good bundle, its because its locally a product with r k 5 section 3. The index formula in cohomology including of course the formula for the numerical index is then derived from this. M is a complex vector bundle together with the structure of a complex manifold on e, such that for any x2mthere exists u3x in mand trivialization. It is not hard to see that this is a rank1 vector bundle over s1. Homogeneous vector bundles raoul bott the annals of. Ebe a cp vector bundle map between cp vector bundles over a cp premanifold with corners x. The primary purpose of this paper is the study of algebraic vector bundles over an elliptic curve defined over an algebraically closed field k. Bplk to the exotic sphere m such that m rk is the smooth structure on the plmanifold e given by smoothing theory, with stable normal bundle m. If the dimension of the vector space is mthen the bundle is often called an mplane bundle. As a corollary, let v be any connection on a vector bundle eonb and let i, vector bundles on x. Pdf vector bundles over normal varieties trivialized by. Let wbe a symplectic vector bundle over a compact manifold zthen there is a natural embedding as a subalgebra. The set of isomorphismclassesofthesegvectorbundlesis denotedby. A vector bundle that can be given an orientation is called an orientable vector bundle.
The dual sequence obtained here sequence 29 was also given by graciasaz and mehta in 7. E x be a vector bundle of rank r over an algebraic variety x. Determinant of a tensor product of two vector bundles. X which is locally trivial, that is, for any x 2 x there exists an open neighborhood u of x and an isomorphism of. In mathematics, an orientation of a real vector bundle is a generalization of an orientation of a vector space. Vector bundles and structure groups a vector bundle over a topological space m or with base space m is, essentially, family of vector spaces continuously parametrized by m. E f is the tensor product bundle, which has rank rk erk f. Nov 12, 2011 let y be a normal and projective variety over an algebraically closed field k and v a vector bundle over y.
Chapter 7 vector bundles louisiana state university. V defined by assigning to each x the subspace e x of v or equivalently the quotient space e x of v is such that1l 2, where11 denotes the sequence on x induced from 1 by. First, i claim that any vector bundle v on xcan be trivialized on the complement of nitely many points. Bass and haboush 2 have shown that the problemhas anaffirmative solution stably. This vector bundle is called the spinor bundle over x, and in fact it is independent on the choice of the line. Vector bundles and connections universiteit utrecht. The most important example is the tangent bundle, a real vector bundle. One also says that the data consisting of an open cover u i. I have reformatted this with narrower margins for a better reading experience on devices like an ipad, but for a paper copy with more standard size margins try printing at 8590 per cent of full size. Sep 27, 2010 vector bundles over normal varieties trivialized by finite morphisms. Vector bundles trivialized by proper morphisms 3 by l we will denote a very ample line bundle on x. In standard bundle terminology, the tangent bundle is an example of a smooth vector bundle of rank nover m.
Such a trivialization is called a holomorphic trivialization. Vector bundles over an elliptic curve 417 it is almost immediate that the map. B, onto b, is called a vector bundle of rank kover bif the following holds. In algebraic geometry, the moduli stack of rankn vector bundles vect n is the stack parametrizing vector bundles or locally free sheaves of rank n over some reasonable spaces it is a smooth algebraic stack of the negative dimension moreover, viewing a rankn vector bundle as a principal bundle, vect n is isomorphic to the classifying stack. Im using the letter m to denote the base space of the vector bundle as a concession to the fact that in most of the applications well be. Vector bundles thus combine topology with linear algebra, and the study of vector bundles could be called linear algebraic topology. The index theorem and formula using the earlier results on ktheory and cohomology the families index theorem of atiyah and singer is proved using a variant of their embedding proof. In the usual terminology w is the universal bundle over the classifying. Equivariant algebraic vector bundles over representations. We also prove that a vector bundle on a homogenous space is trivial if and only if the restriction of the vector bundle to every schubert line is trivial. Definition for x x a topological space, then a topological vector bundle e x e \to x over a topological field k k is called trivial if its total space is the product topological space. Hence any bundle restricted to these will be trivial and essentially the bundle is determined by how you glue the two trivial bundles together on the equator. Xis called the rst stiefelwhitney class, and denoted w 1e.
The comprehensive, creative vectors bundle s of varied. We call mthe base of this bundle, and the 2ndimensional manifold tmitself is called its total space. Counting vector bundles usc dana and david dornsife college. Then there exists a unique connection, called the trivial connection, such that the constant sections i. To construct an ndimensional trivial bundle nover a space x. A smooth map x eis said to be a smooth section of the bundle e,x. This follows from the existence of dimv meromorphic sections of v which are linearly independent at all. Further, the umodule e defines an analytic vector bundle e g x e over x, the author holds a sloan fellowship. The idea for bundles over the spheres could be to say that you can cover the sphere by the two hemispheres, which are contractible.
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