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What is a trivial vector bundle?

What is a trivial vector bundle?

An isomorphism of vector bundles over X of the form. E⟶X×ℝn. is called a trivialization of E. If E admits such an isomorphism, then it is called a trivializable vector bundle.

What is the degree of a line bundle?

Using Theorem 1.1. 1 we can define the degree of a line bundle on C, even when C is not smooth. When L is very ample, one defines deg L = dimk Γ(Z(s), OZ(s)) for a non-zero section s of L. The degree of L, deg (L) is defined by the formula deg (L) = deg (M) − deg (N) where L ∼= M ⊗OC N−1 with M and N very ample on C.

Why are line bundles important?

Similarly, sections of vector bundles and line bundles are a nice way to talk about functions with poles. Meromorphic functions then become simply sections of a line bundle, which is nice because it allows us to avoid having to talk about ∞.

What is the rank of a vector bundle?

If kx is equal to a constant k on all of X, then k is called the rank of the vector bundle, and E is said to be a vector bundle of rank k. Often the definition of a vector bundle includes that the rank is well defined, so that kx is constant.

What is the best definition for bundle?

Bundle is used to pass data between Activities. You can create a bundle, pass it to Intent that starts the activity which then can be used from the destination activity.

Is tangent bundle a manifold?

We want to show that the tangent bundle TM itself is a manifold in a natural way and the projection map π : TM → M is smooth. We will first find candidates for coordinate charts on the tangent bundle TM. They will be constructed out of coordinate charts on M.

What is a positive line bundle?

Positive line bundles In algebraic geometry, positive definite (1,1)-forms arise as curvature forms of ample line bundles (also known as positive line bundles). Let L be a holomorphic Hermitian line bundle on a complex manifold, its complex structure operator.

What is the full meaning of bundle?

1. several objects or a quantity of material gathered or bound together. a bundle of hay. 2. an item, group, or quantity wrapped for carrying; package.

What is the sentence of bundle?

Cynthia noticed him picking at his fingers, a sure sign he was a bundle of nerves. Betsy was a bundle of nerves. Cynthia dropped the bundle on the sofa and held the dress up in front of the woman. He carried a large bundle .

What is mean bundle?

A bundle is a package of things wrapped together. To wrap things together in a compact way is to bundle them. A baby wrapped up in a blanket is a bundle of joy, and if it’s cold outside, bundle up! Bundle comes from the Middle Dutch word for bind, which is what you do when you bundle stuff — you bind it together.

What is the tautological line bundle?

In the case of projective space the tautological bundle is known as the tautological line bundle. The tautological bundle is also called the universal bundle since any vector bundle (over a compact space) is a pullback of the tautological bundle; this is to say a Grassmannian is a classifying space for vector bundles.

Is the twisting sheaf a tautological line bundle?

where x0 is, as usual, viewed as a global section of the twisting sheaf O (1). (In fact, the above isomorphism is part of the usual correspondence between Weil divisors and Cartier divisors.) Finally, the dual of the twisting sheaf corresponds to the tautological line bundle (see below).

What is the inverse map of a trivial bundle?

The inverse map is given as follows: since X is compact, any vector bundle E is a subbundle of a trivial bundle: unique up to homotopy.

What is the standard line bundle in K theory?

In Michael Atiyah ‘s “K-theory”, the tautological line bundle over a complex projective space is called the standard line bundle. The sphere bundle of the standard bundle is usually called the Hopf bundle. (cf. Bott generator .)