Download pdf in this thesis we lay the foundation for rational degree d as an element of z1p by using perfectoid analogue of projective space, and consider power series instead of polynomials. 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. Line bundles of rational degree over perfectoid space. A section of a line bundle is the data of maps g i. Introduction to algebraic geometry, class 21 contents. Line bundles to picard group on projective space youtube.
Sheaf of sections of a line bundle, and the correspondence with line bundles. Let k be a eld and let v be a kvector space of dimension n. The difference between a vector space and the associated af. Vector bundles on projective space university of michigan. In fact, on any smooth projective variety, the dualising sheaf is pre. U i c, satisfying g ipf ijpg jp for points p2u i\u j.
In fact among all compact complex manifolds those which are embeddable in a projective space can be characterized by having \su ciently many divisors, in a sense that we shall make more precise in later sections. Let m be a compact complex space with a line bundle is said to be completely intersected l. Vector bundles over an elliptic curve 417 it is almost immediate that the map. In a nutshell, since projective space has a line bundle whose sections have no common zeroes, and line bundles and sections pull back under maps, having such a line bundle is a necessary condition for a map to projective space. Lines in projective space mathematics stack exchange. On the other hand, our invariants may not be numerical if the line bundle is not big. First, since all roots have the same length, the cotangent bundle of gp identi. There is a noncanonical way to identify sections of a bundle with rational functions having certain prescribed poles explained e. For example, e may be the vector space of real homogeneous polynomialspx,y,z of degree 2 in three variablesx,y,z plus the null polynomial, and a line. Kempf for a vector bundle on projective space to be. Divisors and line bundles jwr wednesday october 23, 2001 8. This formula reduces to weyls character formula, when the projective homogeneous space is a. Line bundles all line bundles discussed in this section are taken to be holomorphic. A technical improvement in the study of seshadri constants is that we no longer require positivity of the line bundles, or that the ambient space is normal.
Vector bundles on projective spaces department of mathematics. Citeseerx line bundles and maps to projective space paul. We shall be concerned with vector bundles over x, i. Two vector bundles over the grasmann g kv are the tautological bundle t. Pn c be a smooth complex subvariety of codimension 2 which is the zero locus of a rank 2 vector bundle. Computer calculations with this character formula reveal the interesting fact that vanishing for ample line bundles on projective homogeneous spaces with nonreduced stabilizers breaks down. For the moment, we only need to say these two line bundles. A family of vector spaces over xis a morphism of varieties e. In this thesis we discuss the theory of vector bundles with real structure on the projective line. Let x s be a projective scheme, o1 a relatively ample line bundle and p a. Hence w 11 n cannot be zero, hence it must be equal to a.
Now if d0is any integer, show that the space of degree d0line bundles is isomorphic to the space of degree dline bundles. We next consider holomorphic line bundles over complex projective space. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases. First introduction to projective toric varieties chapter 1. Naturality properties of chern classes and topological definition. The rst chern class of a complex line bundle is poincar e dual to the vanishing locus of a generic section. In this context, the infinite union of projective spaces direct limit, denoted cp. Vector bundles on projective space takumi murayama december 1, 20 1 preliminaries on vector bundles let xbe a quasiprojective variety over k. A point x 2 cpn corresponds to a line lx line bundles on projective space and grassmannians. In particular, the total space l of a line bundle is also a complex manifold of. The only non trivial part is to show that this map is injective, or eq uival ently, that if e x is a.
This extends classical work by grothendieck classifying complex vector bundles on the projective line. A linear series of degree dand dimension ron xis a pair l. We also discuss connections with other characterizations of projective space. Since this projective space is 1dimensional, we have succeeded in creating the projective line over. It is easy to check that the axioms for a projective space hold. Mhas the following holonomy invariant decomposition.
A characterization of complex projective spaces by sections. Moreover, if eis a vector bundle over x speca, e 7. Pdf complements of hyperplane subbundles in projective. A natural question is to enumerate the set linefx of such bundles. 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. Introduction let lbe an ample line bundle on a smooth projective variety xde. The aim of the study is to apply it to the study of syzygies of discriminants of linear systems on projective space and grassmannians. It is endowed with a very ample invertible sheaf o pv1. Maps to projective space determined by a line bundle. Line bundles and maps to projective space citeseerx. L of line bundles l over toric ambient varieties x. Find materials for this course in the pages linked along the left. 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. Canonical line bundle over a projective bundle mathematics.
Line bundles on projective spaces preliminary draft 3 our description given here. One of the most important line bundles in algebraic geometry is the tautological line bundle on projective space. In topology, the complex projective space plays an important role as a classifying space for complex line bundles. Pn is clearly covered by a bundle map from 1 1 to 1 n. A line bundle l on a projective variety x is ample if some positive. Dec 29, 2012 a very readable introduction to pic group of projective spaces using line bundles. In the presence of a positivity condition, embedding and vanishing. H be the pullback of the line bundle h on p n the dual of the tautological line bundle. We construct natural line bundles opfd on pf for all d. For the moment, we only need to say these two line bundles are duals of each other. Essentially all maps to projective space are of this form. Now, call a scheme sgood if all line bundles on p n 1 s are of the form ok l s for k2z and l s a line bundle on s. As xis projective, it is proper and so iis a closed.
The main result that we will discuss is the following theorem. Quaternionic line bundles over quaternionic projective spaces daciberg l. However, for the purposes of this paper, our terminology should be. Intro to algebraic geometry, problem set 12 line bundles. Since 1 n is a line bundle, the rst axiom for stiefelwhitney classes tells us that the higher classes must be zero. In the usual terminology w is the universal bundle over the classifying. Demailly recently proved a form of the converse to andreottigrauerts theorem for n.
In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a point at infinity. Conversely one may hope that the results obtained by abstract mathematical methods may be of use to the physicists. Vector bundles on projective space takumi murayama december 1, 20 1 preliminaries on vector bundles let xbe a quasi projective variety over k. Lazarsfelds book is an excellent reference on ample line bundles 26.
Z that correspond to the standard line bundles od on projective spaces. October 24, 2018 abstract we study the quantisation of complex. Then one asks whether it is also sufficient, and it is, as above. We start the groundwork by proving weierstrass theorems for perfectoid spaceswhich are analogues of standard weierstrass theorems in complex analysis. One of the basic problems in the theory of vector bundles is to classify all vector bundles over a space x. Lecture notes geometry of manifolds mathematics mit. A canonical treatment of line bundles over general projective spaces. Using linear algebra to classify vector bundles over p1. A canonical treatment of line bundles over general. Our wouldbe projective space has just one line, corresponding to the projection 1, and all the points lie on this line. X, there exists a neighborhood u x such that there is an. A characterization of complex projective spaces by. Hence any subvariety of projective space also has by restriction a line bundle whose sections have no common zeroes. A canonical treatment of line bundles over general projective.
Miles reid, graded rings and varieties in weighted projective space. We can try to understand naive qampleness on projective varieties over any. Our intention is to introduce the isomorphism classes of line bundles on a projective space in three different ways. 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. A canonical treatment of line bundles over general projective spaces 3 2. On the geometry of hypersurfaces of low degrees in the. The projectivization pv of a vector space v over a field k is defined to be the quotient of. The projectivization p v of a vector space v over a field k is defined to be the quotient of v. Cohomology of projective space let us calculate the cohomology of projective space.
Line bundles with partially vanishing cohomology burt totaro ample line bundles are fundamental to algebraic geometry. We study the incidence complex for the line bundle d on the projective line and show it is a resolution of the ideal sheaf of i l d the incidence scheme of d. Examples lines are hyperplanes of p2 and they form a projective space of dimension 2. In particular, we show that vector bundles with real structure can be classi ed in terms of the coroot lattice of gln, similarly to the. One reason to investigate which line bundles are ample is in order to classify algebraic varieties. Let m be a ndimensional compact irreducible complex space with a line bundle l.
How to think of the pullback operation of line bundles. A vector bundle is a family of vector spaces that is locally trivial, i. Characterization of the projective spaces in this paper, a characterization of the projective space will be given. Aug 31, 2011 complements of hy perplane subbundles in projective space bundles over p 1 4 pro of. 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. The degree of the line bundle lis the degree of the map it corresponds to.
Yes, a line in the projective space associated to a vector space is a plane in that vector space. Maps to projective space correspond to a vector space of sections of a line bundle. Vector bundles over an elliptic curve 415 embedded biregularly in some projective space. It is known that there are indecomposable bundles of rank.
If k is the complex field then it has been shown by serre 9 that the algebraic and. The tangent bundle and projective bundle let us give the first. Frobenius seshadri constants and characterizations of. More precisely, this is called the tautological subbundle, and there is also a dual ndimensional bundle called the tautological quotient bundle. Real projective space has a natural line bundle over it, called the tautological bundle. However, for real projective space, the negative degree line bundles do have nontrivial global sections. We sometimes omit algebraic and call it a projective variety. Let a be a regular local ring maximal ideal and x spec a lo. Let a be an a ne space modelled on the fvector space v and let b. In particular, h is a tautological quaternionic line bundle when the base space is a quaternionic projective space hpn. There is a tautological line bundle s pe which restricts on each. This is part of the more general problem to find vector bundles of small rank on large projective spaces. Every nitely generated projective amodule pfor a kx 1x n is free. Complex space, projective space, line bundle, complete intersected.
For any locally principal divisor, one can make a holomorphic line bundle such that. Therefore, it is pretty obvious that it would be important to have a tool to determine these line bundle cohomology classes in a straightforward way, i. This character mirrors the di erences in complex and real line bundles in the holomorphic and real analytic categories, respectively. Second, for 2xp, let l denote the corresponding line bundle on gp. One using taylor maps, incidence schemes, jet bundles and generalized verma modules. 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 previous example is a little restrictive if we require s to be convex, because there are not many convex a nely independent. We denote by g kv the grasmann manifold of kdimensional subspaces of v and by pv g 1v the projective space of v.
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. That the theorem is related to the cohomology of line bundles on the cotangent bundle of a grassmannian goes as follows. There is a tautological bundle over cpn, denoted o 1 for reasons which will soon become clear. Note that there is always a zerosection given by g ip 0 for all i, p2u. Line bundles to picard group on projective space by harpreet bedi. We know of twointeresting pieces of evidence for this conjecture. There exists a space bon such that there is a bijection between e.
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