Zero section vector bundle pdf

On the other hand, it is possible that a vector bundle admits no nonvanishing section. Vanishing locus of a general section of a vector bundle. The set of all sections of eis denoted by e, and the set of all smooth sections of eis denoted by 1e. Are there any general results on when a closed subscheme x of a quasiprojective smooth scheme m can be written as the zeroset of a section of a vector bundle e on m.

Vector bundles and connections universiteit utrecht. Here, a vector bundle eover a nonsingular projective curve xis semistable in the sense of mumfordtakemoto if for all subbundle f of e, there holds f e. Anyway, the the answer to your question is yes this is a special case of the following more general result of bertini type about degeneracy loci of morphism of vector bundles, whose proof can be found in ottavianis book varieta proiettive di codimensione piccola. A section is an abstract characterization of what it means to be a graph. E of e by this is a vector bundle over 0, 1 with fiber e. As donu pointed out, it may happen that the general section has no zeroes. Let e be a holomorphic vector bundle of dimension ttz.

A convenient test is whether or not the intersections with the. U i c, satisfying g ipf ijpg jp for points p2u i\u j. One can see this by noting that if one removes the zero section from ts2, one obtains a space which has the homotopy type of rp3, which has fundamental group z2z. If ehas a nowhere vanishing section, then this implies that eis a direct sum of line bundles and this contradicts stability. It is called the trivial rankkcomplex vector bundle. Over an open set usuch that the vector bundle is trivial, a section is just a collection of nfunctions.

A section of a line bundle is the data of maps g i. The zero section of the tangent bundle tm is the set of zero tangent vectors, z fp. Y is a holomorphic vector bundle e over x, a holomorphic section s of e, and w the zero section of e. The most important example is the tangent bundle, a real vector bundle. Let h be an ample line bundle on x, whose restriction h z to z is generated by global sections. Therefore this section is devoted to the notion of vector bundles. Chapter 7 vector bundles louisiana state university. Prove that the morphism is injective, and that its cokernel that is, the quotient of the target by its image is isomorphic to tpn. The case of quaternionic vector bundles can be handled in the same way. For each plane r in rn define the projection operator ii. Why can we always take the zero section of a vector bundle.

A smooth map x eis said to be a smooth section of the bundle e,x. A section of a tangent vector bundle is a vector field. 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. X in the geometric sense, we can pass to the o xmodule eof sections of. We can use sections to prove in very simple cases that vector bundles are not isomorphic. Ample vector bundles with zero loci of sectional genus two. Vector bundle extensions, sheaf cohomology, and the heterotic. For affine bundles in your sense, as you point out, there is no preferred origin so one cannot just define the zero section using the local charts.

Mar 04, 2011 on double vector bundles can be transferred to o p erations on the corresponding. Conversely, given a vector bundle ein the algebraic sense, the corresponding space. The zero section of a vector bundle is the submanifold of the bundle that consists of all the zero vectors. Remark 2 i a simple example of a section is the zerosection which assigns to each b. Y x \sqcup y is the direct sum of a vector bundle that has rank zero on y y and one that has rank zero on x x.

The set of vector bundles over and morphisms of locally constant rank forms an exact subcategory of the category for any vector bundle and mapping, the induced fibre bundle is endowed with a vector bundle structure such that the morphism is a vector bundle morphism. If mis a smooth manifold and kis a nonnegative integer, then. Vector bundle extensions, sheaf cohomology, and the heterotic standard model volker braun 1,2, yanghui he, burt a. Equivalently, one can consider the pullback bundle.

Hence the zero locus of any non zero section of the rank two vector bundle eis either empty or pure of codimension 2. When is a scheme a zeroset of a section of a vector bundle. The reason for my question was confusion as to the definition of a vector bundle. Let e and b differentiable manifolds and e b a smooth map. A local section of a fiber bundle is a continuous map. Given a space x, we take a real or complex finite dimensional vector space v and. If e 1 and e 2 are semistable in the sense of mumfordtakemoto, then so is e 1 e 2. If there are no other sections, we say that the bundle is. If we regard a section merely as a map between the manifolds m and e, then one answer to the question is provided by the tangent map ts. 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. When is x the fiber product of m e zero section and the other arrow is the section im looking for.

For example, we have seen mis orientable if and only if nt madmits a global nonvanishing section. Computing zeros of sections of vector bundles using. The section smay be viewed as a morphism from the trivial vector bundle pn a1 to v. Fiber bundles do not in general have such global sections consider, for example, the fiber bundle over with fiber. It is not hard to see that this is a rank1 vector bundle over s1. Fix integers 0 sheaf cohomology, and the heterotic standard model volker braun 1,2, yanghui he, burt a. As was explained in the other answers, for vector bundles one requires that the fibers are equipped with a vector space structure, and this immediately gives a continuous section. Zero sets of random sections of vector bundles michael taylor preliminary notes contents 1. H0x,e,generically transverse to the zero section, of a hermitian holomorphic vector bundle eof rank r. In particular, the total space lof a line bundle is also a complex manifold of dimension one higher than that of x, with a morphism l. Miscellany on the zero schemes of sections of vector bundles piotr pragacz institute of mathematics of polish academy of sciences sniadeckich 8, 00956 warszawa, poland p. L2 estimates and vanishing theorems for holomorphic vector bundles equipped with singular hermitian metrics inayama, takahiro, michigan mathematical journal, 2020.

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. Zero sections of any smooth vector bundle is smooth. The set vectpxq of isomorphism classes of complex vector bundles on a topological space x is a homotopy invariant of x. But there are many other interesting vector bundles. If the dimension of the vector space is mthen the bundle is often called an mplane bundle. In fact, by the hairy ball theorem, ts2 has no nowhere.

Miscellany on the zero schemes of sections of vector bundles. A smooth section of the vector bundle e mis a smooth map s. In particular, a vector field on a smooth manifold. This is a 2 dimensional vector space with the point xas its zero vector 0x. E and a vector x 2 txm, what is meant by the directional derivative dsxx. The notion of vector bundle is a basic extension to the geometric domain of the fundamental idea of a vector space.

Roughly speaking, a vector bundle e can be thought of as a manifold m with a vector space e p attached to each point p. But this ignores most of the structure that makes a vector 69. Vector bundles of rank 1 is also called the line bundle. A line bundle is a vector bundle with 1dimensional. Fundamental terminology and results from differential topology as in guillemin and pollack 1974, hirsch 1976, and milnor 1969 are adopted herein.

E be a non zero section whose zero locus cis pure of codimension 2 in x. Show that the quotient vector space i pi2 p is canonically isomorphic to the cotangent space t p m. A kvector bundle over mof rank kconsists of a bundle e m whose. However, since his not commutative, one has to be careful when writing down formulas. Ovrut, and tony pantev2 1 department of physics, 2 department of mathematics david rittenhouse laboratory, university of pennsylvania 209 s. Strictly speaking, a vector bundle is an entire triple e. A bundle over a manifold is trivial if it is simply the cartesian product of. L x be a holomorphic line bunlde x holomorphic section. A complex vector bundle of rank kis also a real vector bundle of rank 2k, but a real vector bundle of rank 2kneed not in general admit a complex structure. For example, sections of the tangent bundle tm n m are vector.