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. If f is not surjective, what is the pushforward going to be away from the image. Pushforward of smooth vector fields physics forums. For instance, to any vector space v we can associate its dual space v. Bismut also showed that under the push forward in topological ktheory superconnections naturally appear even if one starts with just an ordinary connection. The local triviality condition means that for every point. The only two vector bundles with base space a circle and onedimensional. Tn over m, which may in turn be viewed as a section of the vector bundle homtm.
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. Vector bundles thus combine topology with linear algebra, and the study of vector bundles could be called linear algebraic topology. For every local trivialization eu of e, the corresponding local trivialization of the. Each line bundle over a compact space is a pullback of o pn1 for some n. But there are many other interesting vector bundles. If you have fixed coordinates then the jacobian matrix is the matrix of the push forward.
The derivative operator d, also known as pushforward operator, maps d. In fact the push forward operation can be performed in the relative setting for any proper morphisms. From a derivative point of view, the tangent bundle of a manifold tm consists of all the possible. When is the pushforward of a vector bundle still a vector bundle. Strictly speaking, a vector bundle is an entire triple e. 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. A bundle over a manifold is trivial if it is simply the cartesian product of the manifold and a vector space. The pullback of a vector bundle is always a vector bundle. I must admit, however, that the appreciation of this beauty came to physicists only in recent years. 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. The condition you wantneed is that every weil divisor be cartier. Vector bundles can be approached from both an algebraic and a geometric perspective.
A ckvector bundle of rank r over a ckmanifold of dimension n. Tm, one can \ push forward x to a smooth vector eld. But wait a minutedoes this equality even make sense. Notes of talks by tamas hausel in oxford, trinity term, 2010. The q vector space cs is generated by ce, a dominant regular weight. Is the pushforward of a line bundle on the smooth locus of. 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.
Decomposition theorem for semisimple algebraic holonomic d. Qthe pushforward of a vector v2t pm 1 is a vector f v2t qm 2. The neighborhoods u over which the vector bundle looks like a product are called trivializing neighborhoods. I only say this because, in sheafland, you can always pushforward and pullback sheafs when you have a map of the underlying space. Then one can define a pullback bundle over x by setting the fiber over x. Decomposition theorem for semisimple algebraic holonomic dmodules takuro mochizuki rims, kyoto university 2017 november. Considering the frobenius pushforwards of line bundles on toric varieties seems to. You can always pull back a 0,s tensor field, but you cannot always push forward a r,0 tensor field e. 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. Frobenius direct images of line bundles on toric varieties. Greens currents for families of hermitian vector bundles. M \\longrightarrow n between the smooth manifolds m.
When is the pushforward of a vector bundle still a vector. The choice of e n allows to choose in a coherent way the rst chern class by taking the pull back from pn. As the universal bundle is holomorphic in the x direction, we can consider the push forward f. Intuitively this is the object we get by gluing at each point p. Vector bundles in algebraic geometry enrique arrondo notes prepared for the first summer school on complex geometry villarrica, chile 79 december 2010 1. Pull back and push forward two important concepts are those of pull back or pullback or pull back and push forward or push forward or pushforward of maps between manifolds. N denote the bundle projections of the tangent bundles of m and n respectively equivalently see bundle map. The push forward is described in p when gis the general linear group, and in ac for the canonical map gb. 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. However, since his not commutative, one has to be careful when writing down formulas.
Direct sum of vector bundles project gutenberg self. Decomposition theorem for semisimple algebraic holonomic. A quick and dirty introduction to differential geometry. M n and a vector field x on m, it is not usually possible to identify a pushforward of x by. For instance, if f is a finite map and e is a line bundle we obtain that f.
Characteristic classes, homogeneous spaces in luk ecin. You should think of this as a ck choice of vector in t pm for all p. Citeseerx document details isaac councill, lee giles, pradeep teregowda. 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. The case of quaternionic vector bundles can be handled in the same way. If the dimension of the vector space is mthen the bundle is often called an mplane bundle. A bundle over a manifold is trivial if it is simply the cartesian product of. This is, of course, the language of gauge theory andiwillborrow freelyfrommy ownpreliminary lecture notesonthissubject. Introduction suppose an algebraic torus t cr acts on a complex variety xwhich is smooth and complete. 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. A smooth section of tmis called a smooth vector eld on m. 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.
In other words, only assuming that your variety is normal, you get that the push forward of a line bundle i. Jul 11, 2016 vector fields as smooth sections of the tangent bundle. Applying this operation on each bre of a vector bundle e. A smooth real vector bundle v of rank k over a smooth manifold m is a. The convergence of physics with mathematics, especially di. Vectors can always be pushed forward to new vectors, but vector fields may not push forward to vector fields. This construction can also be applied brewise to vector bundles.
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. Where we identify the circle s 1 with r z, there is a continuous map. The thing on the left is a scalar, but the thing on the right is a vector. Does it make sense to pullback or pushforward a mixed tensor. Vector bundles and connections universiteit utrecht. 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. Notes by gergely berczi, michael groechenig and geordie williamson. The pushforward of vector fields is compatible with. Let e be an equivariant complex vector bundle over x. The most important example is the tangent bundle, a real vector bundle.
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. In other words, the derivative of f is just the stretch factor as we go from one copy of r to the other. Under some additional assumptions on v, we are able to. The func torial interpretation of the integration over the virtual fundamen tal classes m.
The ring of functions on a1 is a principal ideal domain, so locally free of nite rank implies free. The equivariant cohomology ring of the moduli space of vector bundles over a riemann surface younghoon kiem. Pushforward of a nontrivial vector bundle by a nontrivial embedding is never a vector bundle it is trivial outside the image of the embedding. There exist various approaches to push forward formulas for. 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. 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.
Pull back and push forward two important concepts are those of pull back or pullback or pull. Chern character of pushforward of vector bundle stack exchange. Let x y be a double ramified cover of complex manifolds. Te, e has a natural secondary vector bundle structure te, p,tm, where p is the push forward of the canonical projection p. 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.
The amount of mathematical sophistication required for a good understanding of modern physics is astounding. We propose to apply the same philosophy to ktheory. From a geometric perspective, recall that an ndimensional vector bundle. We can use the vector space structure on tpm to add vector.
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