Equivalently, e is the vector bundle constructed from eby taking the complex. Algebras of endomorphisms of semistable vector bundles of. Vector bundles are almost always required to be locally trivial, however, which means they are examples of fiber bundles. Wbetween two vector spaces that is compatible with the two vector space operations i t v tv for all 2f and for all v 2v. Review of tensors, manifolds, and vector bundles most of the technical machinery of riemannian geometry is built up us ing tensors. B, onto b, is called a vector bundle of rank kover bif the following holds.
He proves that a vector bundle e over x is indecomposable if and only if ende is a special algebra. The most common case is that of a linear connection on a vector bundle, for which the notion of parallel transport must be linear. It isnaturalin the sensethat it commutes with the induced action on tm of any di. In fact, by the chain rule, one can check that the transition matrices satisfy the transition conditions i. Note on the linear endomorphisms of a vector bundle. Vector bundles in algebraic geometry enrique arrondo notes prepared for the first summer school on complex geometry villarrica, chile 79 december 2010 1. In mathematics, an endomorphism is a morphism from a mathematical object to itself. Since eis selfdual we also get that all slopes are nonnegative, so all slopes are 0. Journal of algebra 123, 414425 1989 algebras of endomorphisms of semistable vector bundles of rank 3 over a riemann surface l.
If eis the complex vector bundle with almost complex structure jwe denote by ethe complex vector bundle with almost complex structure j. Nonliftability of vector bundles to the witt vectors bhargav bhatt we give an example of a vector bundle eon a smooth projective variety xover an algebraically closed eld of characteristic psuch that edoes not extend across the nilpotent thickening x. Further there is an operation of multiplication of the reals on the vectors r. A vector bundle l is known to be orientable for real ltheory if its first two stiefelwhitney classes vanish. Since im not an expert in algebraic geometry i have some difficulties in translating results concering stable algebraic vector bundles over a riemann surface into the complex geometric delbar approach to stable holomorphic vector bundles. Given some context of geometry, then a vector bundle is a collection of vector spaces that varies in a geometric way over a given base space x x. Formality theorem for hochschild cochains of the algebra of. 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. Cohomology and vector bundles university of chicago. The euler class, the thom class and the chern classes are a few. But there are many other interesting vector bundles.
The ktheory of vector bundles with endomorphisms over a. For example, an endomorphism of a vector space v is a linear map f. We denote the algebra of endomorphisms of this bundle by ende. On the twisted endomorphism algebra of a vector bundle on a. Motivation let x,o be a cp premanifold with corners, 0. An algebraic vector bundle of rank r on x is a linear. Intuitively this is the object we get by gluing at each point p.
On the twisted endomorphism algebra of a vector bundle on a curve e. For a trivial vector bundle one can easily prove the desired statement using the formality theorem 5, 6 for hochschild chains of the algebra om of functions on m and the cotrace. If no, what are the usualstandard notations for this. Vector decorative elements ornaments banner floral typography type kit bundle hand drawn sketched frames illustration border shapes corners bursts badge vintage creative market is the worlds marketplace for design. Cohomology and vector bundles corrin clarkson reu 2008 september 28, 2008 abstract vector bundles are a generalization of the cross product of a topological space with a vector space. Characteristic classes assign to the each vector bundle a cohomology class of the base space. The endomorphisms monoid of a homogeneous vector bundle. Endomorphism vector bundle isomorphic to the adjoint bundle. A bundle over a manifold is trivial if it is simply the cartesian product of.
In mathematics, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle. An hermitian structure h in e can be expressed, in terms of a local holomorphic frame field s l9, s r of e, by a. The ktheory of vector bundles with endomorphisms over a scheme. However, since his not commutative, one has to be careful when writing down formulas. V v, and an endomorphism of a group g is a group homomorphism f. A vector bundle f is simple if and only if endf 163. Endomorphisms of stable vector bundles over a riemann surface. Informally speaking, the linear endomorphisms operators are linear transformations or morphisms defined on a vector space and taking values in the same space hence, the definition of a linear morphism def. A kvector bundle over mof rank kconsists of a bundle e m whose. Introduction in this work we describe a noncommutative algebraic approach to such geometrical objects as degenerate poisson structures, symplectic foliations for poisson. The case of quaternionic vector bundles can be handled in the same way. We construct the crossed product of a cxalgebra by an endomorphism, in such a way that the endomorphism itself becomes induced by the bimodule of continuous sections of a vector bundle.
Ballico 1 annali di matematica pura ed applicata volume 179, pages 383 400 2001 cite this article. The most important example is the tangent bundle, a real vector bundle. A connection ron einduces the same connection ron e, but the type decom. In this case it can be read off from the diagram of theorem 3. Thus, any endomorphism of e can be written as a sum of a scalar multiple of the identity and a nilpotent endomorphisms. So to extend the definition of a determinant of an endomorphism of vector spaces to one for an endomorphism of vector bundles, you just need a good natural definition of determinant. Pdf crossed products by endomorphisms, vector bundles. On the noncummutative geometry of the endomorphism algebra. Introduction in 5,6,7 i introduced the concept of einsteinhermitian vector bundle. Also, the vector spaces are usually required to be over the real or complex numbers, in which case the vector bundle is said to be a real or complex vector bundle respectively. In the simplest examples the bundle is a complex line bundle and the higgs.
Isomorphisms math linear algebra d joyce, fall 2015 frequently in mathematics we look at two algebraic structures aand bof the same kind and want to compare them. Ff,f, where f is a vector bundle on x s coherent locally free oox module and. The tangent bundle of x is the real vector bundle tx on x which is given by the transition matrices. Ellenberg department of mathematics, princeton university fine hall, washington road princeton, nj 08544 email. Vector spaces, duals and endomorphisms a real vector space v is a set equipped with an additive operation which is commutative and associative, has a zero element 0 and has an additive inverse vfor any v2v so v is an abelian group under addition. X which is locally trivial, that is, for any x 2 x there exists an open neighborhood u of x and an isomorphism of. A vector space homomorphism that maps v to itself is called an endomorphism of v. Introduction hermitian vector bundles and dirac operators.
The theory of a single endomorphism recall that an endomorphism is a map t. S2 of rank 2 for each k2z or a complex vector bundle of rank 1, since g 01. Noncommutative symplectic geometry of the endomorphism. For instance, we might think theyre really the same thing, but they have different names for their elements. Hermitian yang mills metrics on vector bundles and stability previously, we discussed the application, to innoue surfaces in particular, of the idea that a stable holomorphic vector bundle admits a hermitian yang mills metric, or what is often called a hermitian yang mills metric. In this section we investigate the endomorphism algebra of a. Let e be a holomorphic vector bundle of rank r over a complex manifold m. The tangent bundle tm is indeed a vector bundle, of rank dimm. Note that if jis an almost complex structure then so is j. On the noncummutative geometry of the endomorphism algebra of. Hermitian einstein metrics on vector bundles and stability. An endomorphism that is also an isomorphism is an automorphism. If the dimension of the vector space is mthen the bundle is often called an mplane 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.
Ttm ttm of rank n is an importantfeatureofthe geometryoftm. We study noncommutative generalizations of such notions of the classical symplectic geometry as degenerate poisson structure, poisson submanifold and quotient. In this paper we give some properties of the algebraic and geometric structure of the endomorphisms monoid of a homogeneous vector bundle. Endomorphisms and automorphisms we now specialize to the situation where a vector space homomorphism a. Pdf crossed products by endomorphisms, vector bundles and.
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