Is projective line compact?
, is the topological space of lines passing through the origin 0 in Rn+1. It is a compact, smooth manifold of dimension n, and is a special case Gr(1, Rn+1) of a Grassmannian space.
Is complex projective space Compact?
Complex projective space is compact and connected, being a quotient of a compact, connected space.
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 ∞.
Is the real plane compact?
Lines and planes are not compact, since one can take a set of equally-spaced points in any given direction without approaching any point.
Is projective space orientable?
The projective plane is non-orientable.
Is CP N Compact?
With this topology CPn is compact. This standard atlas gives a complex structure for CPn .
What is complex projective line?
Complex projective line: the Riemann sphere Adding a point at infinity to the complex plane results in a space that is topologically a sphere. It is in constant use in complex analysis, algebraic geometry and complex manifold theory, as the simplest example of a compact Riemann surface.
What is a positive line bundle?
A holomorphic line bundle on a complex manifold is called positive if its curvature differential 2-form is, after multiplication with i=√−1, a positive definite (1,1)-form.
Is the tangent bundle a vector space?
The tangent bundle of the sphere is the union of all these tangent spaces, regarded as a topological bundle of vector space (a vector bundle) over the 2-sphere. A tangent vector on X at x∈X is an element of TxX.
Why is R not compact?
R is neither compact nor sequentially compact. That it is not se- quentially compact follows from the fact that R is unbounded and Heine-Borel. To see that it is not compact, simply notice that the open cover consisting exactly of the sets Un = (−n, n) can have no finite subcover.
Is the union of compact sets compact?
Show that the union of two compact sets is compact, and that the intersection of any number of compact sets is compact. Ans. The union of these subcovers, which is finite, is a subcover for X1 ∪ X2. The intersection of any number of compact sets is a closed subset of any of the sets, and therefore compact.
What is the canonical line bundle over a projective space?
The canonical line bundle over a projective space is sometimes called its “tautological line bundle”. For more see at classifying space. In the following, let k be a star – field, possibly a skew-field k ∈ StarSkewFields.
Why there are no common zeroes in projective space?
I.e. by its very definition, projective space carries a tautological line bundle, whose dual bundle has as sections the linear coordinates. These sections have no common zeroes because the hyperplanes have no common points. Hence any subvariety of projective space also has by restriction a line bundle whose sections have no common zeroes.
What is the tautological K-line bundle over the projective space?
The tautological k-line bundle over the projective space kPn is the following vertical bundle map: regarded with the right k × – group action: k * is k equipped with the right k × -action by inverse multiplication from the left: (equivalently, (3) and (4) are left actions of the opposite group (k ×)op)
What is a tautological line bundle?
The line bundle (1) is “tautological” in the sense that its fiber over a point labeled [v] – which may be regarded as the name of the line spanned by the vector v ∈ kn + 1 – consists of all the points v ⋅ z on that line – as made explicit by the horizontal map in (1).