Push forward of vector bundle pdf

Universal gysin formulas for flag bundles 3 formulae for grassman bundles, ii, kaji and terasoma prove a similar formula to that of ilori in type a in the particular case of full. The purpose of this note is to study the properties of holomorphic rank 2 vector bundles on y arising as push forward of line bundles on. The most important example is the tangent bundle, a real vector bundle. The derivative operator d, also known as pushforward operator, maps d. You should think of this as a ck choice of vector in t pm for all p. We will use the transition function point of view, thinking of a vector bundle over p1 as glued together from two vector bundles over a1 by a transition function. Tm, one can \ push forward x to a smooth vector eld. Motivated by the study of the hitchin fibration, they study not the pushforward of ox itself but the pushforwards of general line bundles on x. The equivariant cohomology ring of the moduli space of vector bundles over a riemann surface younghoon kiem. A smooth section of tmis called a smooth vector eld on m.

I only say this because, in sheafland, you can always pushforward and pullback sheafs when you have a map of the underlying space. Introduction suppose an algebraic torus t cr acts on a complex variety xwhich is smooth and complete. The pullback of a vector bundle is always a vector bundle. Does it make sense to pullback or pushforward a mixed tensor. 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. Let e be an equivariant complex vector bundle over x. In fact the push forward operation can be performed in the relative setting for any proper morphisms.

The convergence of physics with mathematics, especially di. The push forward is described in p when gis the general linear group, and in ac for the canonical map gb. Tn over m, which may in turn be viewed as a section of the vector bundle homtm. We prove that, up to twist, every vector bundle on a smooth projective curve arises from the direct image of the structure sheaf of a smooth, connected branched cover. M \\longrightarrow n between the smooth manifolds m. This is, of course, the language of gauge theory andiwillborrow freelyfrommy ownpreliminary lecture notesonthissubject. The func torial interpretation of the integration over the virtual fundamen tal classes m. Te, e has a natural secondary vector bundle structure te, p,tm, where p is the push forward of the canonical projection p. A bundle over a manifold is trivial if it is simply the cartesian product of the manifold and a vector space. You can always pull back a 0,s tensor field, but you cannot always push forward a r,0 tensor field e. In other words, only assuming that your variety is normal, you get that the push forward of a line bundle i. Vector bundles thus combine topology with linear algebra, and the study of vector bundles could be called linear algebraic topology. I must admit, however, that the appreciation of this beauty came to physicists only in recent years.

Notes of talks by tamas hausel in oxford, trinity term, 2010. Chern character of pushforward of vector bundle stack exchange. 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. 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. The condition you wantneed is that every weil divisor be cartier. We can use the vector space structure on tpm to add vector. X y be a proper morphism of noetherian schemes, with y reduced and connected, and e a coherent sheaf on x, flat over y. We establish the thom isomorphism in twisted ktheory for any real vector bundle and develop the push forward map in twisted ktheory for any di. The amount of mathematical sophistication required for a good understanding of modern physics is astounding. Notes by gergely berczi, michael groechenig and geordie williamson. N denote the bundle projections of the tangent bundles of m and n respectively equivalently see bundle map. If f is not surjective, what is the pushforward going to be away from the image. The pushforward of vector fields is compatible with. Under some additional assumptions on v, we are able to.

The case of quaternionic vector bundles can be handled in the same way. For any smooth vector bundle e, p, m the total space te of its tangent bundle te. Greens currents for families of hermitian vector bundles. M n and a vector field x on m, it is not usually possible to identify a pushforward of x by. Decomposition theorem for semisimple algebraic holonomic. Frobenius direct images of line bundles on toric varieties. Characteristic classes, homogeneous spaces in luk ecin. N denote the bundle projections of the tangent bundles of m and n respectively. U of the universal bundle, which is a bundle of rank 2g by riemannroch. In mathematics, particularly differential topology, the secondary vector bundle structure refers to the natural vector bundle structure te, p. Considering the frobenius pushforwards of line bundles on toric varieties seems to. The thing on the left is a scalar, but the thing on the right is a vector. A quick and dirty introduction to differential geometry. The purpose of this note is to study the properties of holomorphic rank2 vector bundles on y arising as push forward of line bundles on x, extending some results of s, fr to the nonprojective case.

Citeseerx document details isaac councill, lee giles, pradeep teregowda. Decomposition theorem for semisimple algebraic holonomic dmodules takuro mochizuki rims, kyoto university 2017 november. A bundle over a manifold is trivial if it is simply the cartesian product of. For every local trivialization eu of e, the corresponding local trivialization of the. Intuitively this is the object we get by gluing at each point p. However, since his not commutative, one has to be careful when writing down formulas. When is the pushforward of a vector bundle still a vector bundle. Vector bundles can be approached from both an algebraic and a geometric perspective. We propose to apply the same philosophy to ktheory.

Applying this operation on each bre of a vector bundle e. If the dimension of the vector space is mthen the bundle is often called an mplane bundle. The purpose of this note is to study the properties of holomorphic rank 2 vector bundles on y arising as push forward of line bundles on x, extending some. But wait a minutedoes this equality even make sense. Each line bundle over a compact space is a pullback of o pn1 for some n. Jetcalculuspushforwardtotalvector push forward a total vector field by a transformation calling sequences pushforwardtotalvector parameters a transformation between two jet spaces description examples description let and be two fiber bundles. For instance, to any vector space v we can associate its dual space v. This construction can also be applied brewise to vector bundles. Pull back and push forward two important concepts are those of pull back or pullback or pull. The neighborhoods u over which the vector bundle looks like a product are called trivializing neighborhoods.

Pushforward of a nontrivial vector bundle by a nontrivial embedding is never a vector bundle it is trivial outside the image of the embedding. University of naples federico ii italy department of structural engineering continuum mechanics on manifolds giovanni romano with the collaboration of. Calculate the jacobian and then multiply it by the appropriate vector. Pushforward of smooth vector fields physics forums. For instance, if f is a finite map and e is a line bundle we obtain that f. In other words, the derivative of f is just the stretch factor as we go from one copy of r to the other. Vector bundles in algebraic geometry enrique arrondo notes prepared for the first summer school on complex geometry villarrica, chile 79 december 2010 1. Principal bundles, chern classes, and abelian instantons hot network questions since every human has a different dna different combinations of c, g, a, t what does it mean to have the genome done.

From a geometric perspective, recall that an ndimensional vector bundle. Is the pushforward of a line bundle on the smooth locus of. There exist various approaches to push forward formulas for. From a derivative point of view, the tangent bundle of a manifold tm consists of all the possible.

As the universal bundle is holomorphic in the x direction, we can consider the push forward f. If you have fixed coordinates then the jacobian matrix is the matrix of the push forward. Decomposition theorem for semisimple algebraic holonomic d. Let x y be a double ramified cover of complex manifolds. Then one can define a pullback bundle over x by setting the fiber over x. Jul 11, 2016 vector fields as smooth sections of the tangent bundle. The q vector space cs is generated by ce, a dominant regular weight. Qthe pushforward of a vector v2t pm 1 is a vector f v2t qm 2. The ring of functions on a1 is a principal ideal domain, so locally free of nite rank implies free. Decomposition theorem for semisimple algebraic holonomic dmodules. The choice of e n allows to choose in a coherent way the rst chern class by taking the pull back from pn. Vectors can always be pushed forward to new vectors, but vector fields may not push forward to vector fields. The pushforward of a nontrivial vector bundle by a nontrivial embedding is never a vector bundle it is trivial outside the image of the embedding.

Vector bundles and connections universiteit utrecht. The local triviality condition means that for every point. A ckvector bundle of rank r over a ckmanifold of dimension n. Direct sum of vector bundles project gutenberg self. But there are many other interesting vector bundles. When is the pushforward of a vector bundle still a vector. Strictly speaking, a vector bundle is an entire triple e. A smooth real vector bundle v of rank k over a smooth manifold m is a. Where we identify the circle s 1 with r z, there is a continuous map. The only two vector bundles with base space a circle and onedimensional.