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