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. B, onto b, is called a vector bundle of rank kover bif the following holds. An algebraic vector bundle of rank r on x is a linear. A vector bundle that can be given an orientation is called an orientable vector bundle. One also says that the data consisting of an open cover u i. The basic invariant of an oriented bundle is the euler class. A vector bundle of rank 1 is usually called a line bundle. Accordingly, the local triviality condition in the definition of topological vector bundles def. 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. Homogeneous vector bundles raoul bott the annals of. The notion of vector bundle is a basic extension to the geometric domain of the fundamental idea of a vector space. Sorry, we are unable to provide the full text but you may find it at the following locations.
Induced bundles and vector bundles induced bundles. Vector bundles over normal varieties trivialized by finite. In standard bundle terminology, the tangent bundle is an example of a smooth vector bundle of rank nover m. H is an isomorphism of vector bundles satisfying that 4. The set of isomorphismclassesofthesegvectorbundlesis denotedby.
Strictly speaking, a vector bundle is an entire triple e. Equivariant algebraic vector bundles over representations. 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. Nov 12, 2011 let y be a normal and projective variety over an algebraically closed field k and v a vector bundle over y.
Vector bundles thus combine topology with linear algebra, and the study of vector bundles could be called linear algebraic topology. The comprehensive, creative vectors bundle s of varied. First, i claim that any vector bundle v on xcan be trivialized on the complement of nitely many points. Degrees of vector bundles are always taken with respect to the polarization l. Sep 27, 2010 vector bundles over normal varieties trivialized by finite morphisms.
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. Such a trivialization is called a holomorphic trivialization. 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. We call mthe base of this bundle, and the 2ndimensional manifold tmitself is called its total space. As a corollary, let v be any connection on a vector bundle eonb and let i, vector bundles on x. To construct an ndimensional trivial bundle nover a space x. The vintage vector bundle is a great pack of handmade resources. Sections section the projection e m sends an entire r k to a point. E m has zero curvature, if and only if, e can be trivialized locally by sections s 1.
The index formula in cohomology including of course the formula for the numerical index is then derived from this. The dual sequence obtained here sequence 29 was also given by graciasaz and mehta in 7. A kvector bundle over mof rank kconsists of a bundle e m whose. X be a vector bundle of rank r over an algebraic variety x. 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. The idea for bundles over the spheres could be to say that you can cover the sphere by the two hemispheres, which are contractible. Determinant of a tensor product of two vector bundles. E 2 of trivialized vector bundles satisfying the condition. 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. E f is the tensor product bundle, which has rank rk erk f. A vector bundle for which a trivialization exists is called trivializable accordingly, the local triviality condition in the definition of topological vector bundles def. 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. 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.
Vector bundles over an elliptic curve 417 it is almost immediate that the map. Then there exists a unique connection, called the trivial connection, such that the constant sections i. We prove that if there exist a kscheme x and a finite surjective morphism g. Induced bundles and vector bundles equivalence classes of line bundles and cohomology classes in h1x. 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. And if you have a good bundle, its because its locally a product with r k 5 section 3. Counting vector bundles usc dana and david dornsife college. Let wbe a symplectic vector bundle over a compact manifold zthen there is a natural embedding as a subalgebra. In the case there is no ambiguity about the base, we will denote a vector bundle by efor short. Vector bundles of rank 1 is also called the line bundle. If x is a smooth, connected and projective variety over an algebraically closed. Suppose that qis nondegenerate on the generic bre of e. A smooth map x eis said to be a smooth section of the bundle e,x. Motivation let x,o be a cp premanifold with corners, 0.
E x be a vector bundle of rank r over an algebraic variety x. Subbundles and quotient bundles motivation example n. X y that trivializes v then v is essentially finite. The only two vector bundles with base space a circle and onedimensional. 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. 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. The multiplication that is, cup product by the euler class of an oriented bundle gives rise to a gysin sequence. It is a rst example of a characteristic class of a ber bundle. Given a space x, we take a real or complex finite dimensional vector space v and. E is obtained from the cartesian product g x e by identifying gu x e with g x u. A bundle over a manifold is trivial if it is simply the cartesian product of.
This follows from the existence of dimv meromorphic sections of v which are linearly independent at all. Vector bundles over normal varieties trivialized by finite morphisms. 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. Ebe a cp vector bundle map between cp vector bundles over a cp premanifold with corners x. Xis called the rst stiefelwhitney class, and denoted w 1e. 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. L with values in some line bundle l over b which is not identically zero on any bre. This new vector bundle fe is called the \associated vector bundle to e, given by the functor f. In mathematics, an orientation of a real vector bundle is a generalization of an orientation of a vector space. A connection on any vector bundle gives a way of parallel transporting sections along curves. Chapter 7 vector bundles louisiana state university. 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. 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.
In the usual terminology w is the universal bundle over the classifying. 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. A vector bundle for which a trivialization exists is called trivializable. But there are many other interesting vector bundles. The primary purpose of this paper is the study of algebraic vector bundles over an elliptic curve defined over an algebraically closed field k. Construct a ranktwo vector bundle over the smooth quadric x. X which is locally trivial, that is, for any x 2 x there exists an open neighborhood u of x and an isomorphism of.
Vector bundles trivialized by proper morphisms 3 by l we will denote a very ample line bundle on x. A vector bundle together with an orientation is called an oriented bundle. Let y be a normal and projective variety over an algebraically closed field k and v a vector bundle over y. 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 most important example is the tangent bundle, a real vector bundle. U ck that is a biholomorphic map of complex manifolds. 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. If the dimension of the vector space is mthen the bundle is often called an mplane bundle. Further, the umodule e defines an analytic vector bundle e g x e over x, the author holds a sloan fellowship. For any smooth manifold m, e m rk is a trivial bundle over m. 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 vector bundle of rank rover xis said to be trivial if there exists a global bundle chart e x. It is not hard to see that this is a rank1 vector bundle over s1. Pdf vector bundles over normal varieties trivialized by. A trivial bundle can be trivialized over its entire base space.
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