Bundles over quantum realweighted projective spaces. To get examples with more interesting higgs fields we can look at projective spaces. Line bundles on projective space daniel litt we wish to show that any line bundle over pn k is isomorphic to om for some m. The only non trivial part is to show that this map is injective, or eq uival ently, that if e x is a. Fix an embedding k cv for each place v,wherecv is the completion of the algebraic closure of kv. A canonical treatment of line bundles over general projective spaces.
Vector bundles over an elliptic curve 415 embedded biregularly in some projective space. The tautological line bundle and the hyperplane bundle are exactly the two generators of the picard group of the projective space. Recall that an adelic metric over lis a cvnorm v over the. Line bundles and maps to projective space citeseerx. Here bundle simply means a local product with the indicated. In particular, the total space l of a line bundle is also a complex manifold of dimension. Extendibility of negative vector bundles over the complex.
Every line bundle has a trivial section, namely the zero section sending the points of xto. At about the same time atiyah and singer as2 made the connection between determinant line bundles and anomalies in physics. The tangent bundle and projective bundle let us give the first. Here is another interesting vector bundle the so called tautological line bundle over the projective space rpn. Vector bundles over an elliptic curve 417 it is almost immediate that the map. With the transition functions gof a line bundle, we can patch the local products u. 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.
A section of a line bundle is the data of maps g i. In this paper, a characterization of the projective space will be given. H be the pullback of the line bundle h on pn the dual of the. Let a be an a ne space modelled on the fvector space v and let b. Introduction to algebraic geometry, class 21 contents. X, there exists an open neighborhood u of xsuch that p.
There is a tautological bundle over cpn, denoted o 1 for reasons which will soon become clear. Generalized holomorphic bundles and the bfield action. The canonical line bundle over a projective space is sometimes called its. Vector bundles and connections universiteit utrecht. Unfortunately threefold intersections cause a conceptual block. Its holomorphic tangent bundle will be denoted tcpn. Using linear algebra to classify vector bundles over p1. Hence any subvariety of projective space also has by restriction a line bundle whose sections have no common zeroes. We shall be concerned with vector bundles over x, i.
Weusethelanguageof adelicmetrizedlinebundles byzhangzh1,zh2. The universal line bundle, o pv1 on the projective space pv will be obtained via the blowup. For a given line bundle lon a smooth variety x, we will denote the set of global sections by h0x. We denote by g kv the grasmann manifold of kdimensional subspaces of v and by pv g 1v the projective space of v.
Canonical line bundle over a projective bundle mathematics. Klaus wirthmuller, section 2 of vector bundles and ktheory, 2012 pdf. The following theorem is a special case of theorem 3. Vector bundles on projective space takumi murayama december 1, 20 1 preliminaries on vector bundles let xbe a quasiprojective variety over k. Examples lines are hyperplanes of p2 and they form a projective space of dimension 2. Essentially all maps to projective space are of this form.
Divisors and line bundles jwr wednesday october 23, 2001 8. More precisely, this is called the tautological subbundle, and there is also a dual ndimensional bundle called the tautological quotient bundle. Let m be a compact complex space with a line bundle is said to be completely intersected l. For the moment, we only need to say these two line bundles are duals of each other. It is easy to check that the axioms for a projective space hold. Given a lie group g, a principal g bundle over a space bcan be viewed as a parameterized family of spaces f x, each with a free, transitive action of gso in particular each f x is homeomorphic to g. A canonical treatment of line bundles over general projective. Real projective space has a natural line bundle over it, called the tautological bundle. Algebraic topology of real projective spaces homotopy groups. M with respect to a line bundle l, provided that complex subspace v 1. By schwarzenbergers property, a complex vector bundle of dimension t over the complex projective space cp nis extendible to cp. Line bundle on a complex manifold with classical topology. Let us give the first nontrivial example of a vector bundle on pn.
Miles reid, graded rings and varieties in weighted projective space. Introduction let lbe an ample line bundle on a smooth projective variety xde. The following line bundles over cpn will be considered. However, an ample line bundle l on x or more precisely some power of l determines an embedding of x into a projective space. When i was a graduate student, my advisor phillip griffiths told me that the grothendieck splitting theorem was equivalent to the kronecker pencil lemma, which gives a normal form for a 2dimensional space of rectangular matrices. We establish that the isomorphy type as an abstract algebraic variety of the complement of an ample hyperplane subbundle h of a projective space bundle of rank r1 over the projective line. In michael atiyahs ktheory, the tautological line bundle over a complex projective space is called the standard line bundle. In this paper, we will prove the borelweil theorem, which relates. These similarities motivate and guide generalizations of well known theorems. Our wouldbe projective space has just one line, corresponding to the projection 1, and all the points lie on this line.
One of the most important line bundles in algebraic geometry is the tautological line bundle on projective space. We also discuss connections with other characterizations of projective space. In this paper, we show some conditions for a negative multiple of a complex line bundle over cpn to be. This is a vector bundle with a onedimensional fibre, and a twist as. In the usual terminology w is the universal bundle over the classifying. A characterization of complex projective spaces by. Divisors and line bundles department of mathematics. The sphere bundle of the standard bundle is usually called the hopf bundle. Let m be a ndimensional compact irreducible complex space with a line bundle l. V defined by assigning to each x the subspace e x of v or equivalently the quotient space e x of v is such that1l 2, where11 denotes the sequence on x induced from 1 by. We next consider holomorphic line bundles over complex projective space. Line bundles over flag varieties ng hoi hei janson abstract. Then the meromorphic sections of the trivial line bundle are just rational functions.
A lie algebra is a vector space g over c with a bilinear form. The projectivization p v of a vector space v over a field k is defined to be the quotient of v. Demailly in dem92, 6 introduced the seshadriconstant. Two vector bundles over the grasmann g kv are the tautological bundle t. Principal u1bundles over quantum real weighted projective spaces are constructed.
Does there exist an indecomposable vector bundle of rank 2 on. For a positive integer n, let rp n denote the real projective space of dimension n. A canonical treatment of line bundles over general. A vector bundle is a family of vector spaces that is locally trivial, i. A characterization of complex projective spaces by sections. Vsuch that our given projective space over a point m. Notes on principal bundles and classifying spaces stephen a. Aug 31, 2011 we establish that the isomorphy type as an abstract algebraic variety of the complement of an ample hyperplane sub bundle h of a projective space bundle of rank r1 over the projective line.
Vector bundles on projective space takumi murayama december 1, 20 1 preliminaries on vector bundles let xbe a quasi projective variety over k. A canonical treatment of line bundles over general projective spaces 3 2. A morphism to projective space is given by a line bundle and a choice of. The projectivization pv of a vector space v over a field k is defined to be the quotient of. Vector bundles on projective space university of michigan. Here a holomorphic line bundle is called positive if and only if it has a hermitian structure for which the curvature is a kahler form. Representation as the pullback of the circle bundle of the tautological line bundle over complex projective space of some dimension to an. In the simplest case of bundles on the projective line we show that. Since this projective space is 1dimensional, we have succeeded in creating the projective line over. Moreover a point of projective space is determined by the set of hyperplanes through it, so any subvariety is determined by the restricted line bundle, since each point is recovered from the set of sections vanishing on it. Characterization of the projective spaces in this paper, a characterization of the projective space will be given. The projective space comes equipped with two line bundles, called the universal line bundle and the hyperplane bundle, denoted by o pv1 and o pv1 respectively. The canonical line bundle over a projective space is sometimes called its tautological line bundle.
We discuss the tautological line bundle as a topological vector bundle. There is a tautological bundle over cp n, denoted o1 for reasons which will soon. With the transition functions gof a line bundle, we can patch the local products uc to get a complex manifold, the total space of the line bundle. Complex space, projective space, line bundle, complete intersected. Mitchell august 2001 1 introduction consider a real nplane bundle. A family of vector spaces over xis a morphism of varieties e.
Frobenius seshadri constants and characterizations of. Stable extendibility of vector bundles over real projective. U i c, satisfying g ipf ijpg jp for points p2u i\u j. Note that there is always a zerosection given by g ip 0 for all i, p2u. Maps to projective space determined by a line bundle. If k is the complex field then it has been shown by serre 9 that the algebraic and. In this section we want to touch few elementary, yet important details on vec tor spaces, affine spaces and projective spaces, and their bundle theoretic analogs. Pdf complements of hyperplane subbundles in projective.
The projectivization pv of a vector space v over a field k is defined to be the quotient of by the action of the multiplicative group k. Michiel hazewinkel and clyde martin, a short elementary proof of grothendiecks theorem on algebraic vectorbundles over the projective line, journal of pure and applied algebra 25 1982, pp. We sometimes omit algebraic and call it a projective variety. Kuroki on projective bundles over small covers bundle of the whitney sum of some line bundles over real projective space, and he found counterexamples to cohomological rigidity in height 2 generalized real bott manifolds. It follows that projective bundles over small covers are not determined by the cohomology ring only.
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