Projectivized vector bundle pdf

The noncommutative analog of the projectivized vector bundle pxe is discussed, and conditions are described for this construction to extend to the noncommutative situation. We have developed the notion of a cp vector bundle over xas a certain kind of cp mapping e xthat is roughly speaking a cp. We prove that the cox ring of the projectivization pe of a rank two toric vector bundle e, over a toric variety x, is a finitely generated kalgebra. Request pdf on the existence of a weak zariski decomposition on projectivized vector bundles a pseudoeffective divisor is said to have a weak zariski decomposition if it can be, up to a. B, onto b, is called a vector bundle of rank kover bif the following holds. For basic results about nef and ample vector bundles, as well as the big vector bundles and qtwisted vector bundles discussed below, see laz, chapter 6. For these projectivized bundles, we give generators for the cone of effective divisors and a presentation of the cox ring as a polynomial algebra over the. Pdf line bundles for which a projectivized jet bundle is. However, for the purposes of this paper, our terminology should be. We prefer to think of sections as \vector valued maps on m, which can be added and multiplied by scalars, and wed like to think of the directional derivative dsxx as something which respects this linear structure. We freely use the additive notation for line bundles. Introduction let e x be a rank r holomorphic vector bundle over a compact complex manifold x of dimension n, and let pe x be the projectivized bundle of lines of e.

Homogeneous vector bundles raoul bott the annals of. Pdf line bundles for which a projectivized jet bundle is a. Swani serre 9, 50 has shown that there is a onetoone correspondence between algebraic vector bundles over an affine variety and finitely generated projective mo. We generalize a construction of hitchin to prove that, given any compact kahler manifold m with positive holomorphic sectional curvature and any holomorphic vector bundle e over m, the projec.

Line bundles on projective spaces preliminary draft 3 our description given here. 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. Cox rings and pseudoeffective cones on projectivized toric. The primary purpose of this paper is the study of algebraic vector bundles over an elliptic curve defined over an algebraically closed field k. We say a vector bundle is spanned if the global sections generate each ber of the bundle. Tve recall the iden tity in question for finite groups. If f is a vector bundle and l a line bundle on x then pf. We study projectivizations of a special class of toric vector bundles that includes cotangent bundles, whose associated klyachko. We study projectivizations of a special class of toric vector bundles that includes cotangent bundles whose associated klyachko. As a consequence, pe is a mori dream space if the toric variety x is projective and simplicial. The work in this dissertation was motivated by several known results and open. Our purpose is to study several of the positivity notions studied for vector bundles with some notions of asymptotic base loci that can be defined on the variety itself, rather than on the projectivization of the given vector bundle. An equivariant or toric vector bundle e on x is a locally free sheaf of.

The only two vector bundles with base space a circle and onedimensional. On the existence of a weak zariski decomposition on projectivized vector bundles article in geometriae dedicata 1791. The tangent bundle tm is indeed a vector bundle, of rank dimm. But there are many other interesting vector bundles. It is well known that a complex finsler structure f on a holomorphic vector bundle e over a compact kahler manifold m. Angelynn alvarez, gordon heier, fangyang zheng submitted on 27 jun 2016. Mar 31, 2015 the aim of this note is to shed some light on the relationships among some notions of positivity for vector bundles that arose in recent decades.

Then the gmodules always left are in an obvious way also limodules. Keywords cox ring pseudoeffective cone toric vector bundle mori dream space torus quotient. Projectivized rank two toric vector bundles are mori dream spaces jos e luis gonz alez abstract. Vector bundles and connections universiteit utrecht. Every vector bundle over a variety x gives a projective bundle by taking the projective spaces of the fibers, but not all projective bundles arise in this way. A kvector bundle over mof rank kconsists of a bundle e m whose. Line bundles for which a projectivized jet bundle is a product article pdf available in proceedings of the american mathematical society 1296 july 1998 with 29 reads how we measure reads. Using linear algebra to classify vector bundles over p1. Vector bundles thus combine topology with linear algebra, and the study of vector bundles could be called linear algebraic topology. Line bundles for which a projectivized jet bundle is a product article pdf available in proceedings of the american mathematical society 1296. We prove that the cox ring of the projectivization pe of a rank two toric vector bundle e, over a toric variety x, is a nitely generated kalgebra. Moreover, if eis a vector bundle over x speca, e 7.

In this paper we analyze the more general setting where lis a line bundle on x with no hypothesis on its positivity and j kl l h,withha line bundle on x. Choose from over a million free vectors, clipart graphics, vector art images, design templates, and illustrations created by artists worldwide. Conversely there is a functor which con verts umodules into gmodules. A bundle over a manifold is trivial if it is simply the cartesian product of. From this perspective the answer is ambiguous, unless e happens to. Vector bundles on projective space university of michigan. Introduction projectivizations of toric vector bundles over complete toric varieties are a large.

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. Bundle of female doctor, physician or surgeon standing in different postures. It is wellknown that a line bundle on a complete toric variety is nef or ample if and. We construct noncommutative ruled surfaces, over a smooth curve x, as noncommutative projectivizations of rank 2 locally free oxbimodules. On projectivized vector bundles and positive holomorphic sectional curvature authors. I have reformatted this with narrower margins for a better reading experience on devices like an ipad, but for a paper copy with more standard size margins try printing at 8590 per cent of full size. In fact, tensoring f with l just multiplies the transition matrices y i. We also discuss an explicit curvature pinching constant for projectivized rank 2 vector bundles over cpn formally known as the hirzebruch surfaces. We often use the same symbols for a vector bundle and its associated locally free sheaf of germs of holomorphic sections. S2 of rank 2 for each k2z or a complex vector bundle of rank 1, since g 01. A vector bundle eon xis nef or ample if the line bundle o1 on pe is nef or ample, respectively. Motivation let x,o be a cp premanifold with corners, 0. The interest of the elliptic curve lies in the fact that it provides the first nontrivial case, grothendieck 6 having shown that for a rational curve every vector bundle is a direct sum of line. Linear algebra operations on vector bundles motivation.

If the dimension of the vector space is mthen the bundle is often called an mplane bundle. The difference between a vector space and the associated af. The most important example is the tangent bundle, a real vector bundle. Vector bundles in algebraic geometry enrique arrondo notes prepared for the first summer school on complex geometry villarrica, chile 79 december 2010 1. On the existence of a weak zariski decomposition on. On positivity and base loci of vector bundles springerlink. Cox rings and pseudoeffective cones on projectivized toric vector bundles jose gonzalezuniversity of michigan toric vector bundles a toric variety is a normal variety xthat contains a torus t. On projectivized vector bundles and positive holomorphic sectional curvature angelynn alvarez, gordon heier, and fangyang zheng abstract.