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# Vector bundles on complex projective spaces by Christian Okonek

Written in English

## Subjects:

• Vector bundles,
• Algebraic Geometry,
• Complex manifolds,
• Projective spaces

Edition Notes

## Book details

Classifications The Physical Object Statement Christian Okonek, Michael Schneider, Heinz Spindler ; with an appendix by S.I. Gelfand Series Modern Birkhäuser classics Contributions Schneider, Michael, 1942 May 18-, Spindler, Heinz, 1947- LC Classifications QA564 .O57 2011 Pagination viii, 239 p. : Number of Pages 239 Open Library OL25269880M ISBN 10 9783034801508 LC Control Number 2011930254

It is intended to serve as an introduction to the topical question of classification of holomorphic vector bundles on complex projective spaces, and can easily be read by students with a basic knowledge of analytic or algebraic geometry. Short supplementary sections describe more advanced topics, further results, and unsolved problems.

It is intended to serve as an introduction to the topical question of classification of holomorphic vector bundles on complex projective spaces, and can easily be read by students with a basic knowledge of analytic or algebraic geometry.

Short supplementary sections Vector bundles on complex projective spaces book more advanced topics, further results, and unsolved by: About this book Introduction It is intended to serve as an introduction to the topical question of classification of holomorphic vector bundles on complex projective spaces, and can easily be read by students with a basic knowledge of analytic or algebraic geometry.

The book is intended for students who have a basic knowledge of analytic and (or) algebraic geometry. Some funda­ mental results from these fields are summarized at the beginning.

One of the authors gave a survey in the Seminaire Bourbaki on the current state of. It is intended to serve as an introduction to the topical question of classification of holomorphic vector bundles on complex projective spaces, and can easily be read by students with a basic knowledge of analytic or algebraic geometry.

Short supplementary sections describe more advanced topics, further results, and unsolved problems.5/5(1).

Additional Physical Format: Online version: Okonek, Christian, Vector bundles on complex projective spaces. Boston: Birkhauser, © (OCoLC) Vector Bundles on Complex Projective Spaces Christian Okonek, Michael Schneider, Heinz The book is intended for students who have a basic knowledge of analytic and (or) algebraic geometry.

Some funda­ mental results from these fields are summarized at the beginning. One of the authors gave a survey in the Seminaire Bourbaki on the. Vector Bundles on Complex Projective Spaces by Christian Okonek,available at Book Depository with free delivery worldwide.

Book reviews. VECTOR BUNDLES ON COMPLEX PROJECTIVE SPACES (Progress in Mathematics, Vol. 3) Cited by: 3. Vector Bundles on Projective Space Takumi Murayama December 1, 1 Preliminaries on vector bundles Let Xbe a (quasi-projective) variety over k.

We follow [Sha13, Chap. 6, x]. De nition. A family of vector spaces over Xis a morphism of varieties ˇ: E!X such that for each x2X, the ber E x:= ˇ 1(x) is isomorphic to a vector space Ar k(x)File Size: KB.

Vector Bundles on Complex Projective Spaces (Progress in Mathematics) $Only 2 left in stock (more on the way). Enter your mobile number or email address below and we'll send you a link to download the free Kindle App. Then you can start reading Kindle books on your smartphone, tablet, or computer - no Kindle device required. The statement and proof of the Kronecker pencil lemma can be found in Gantmacher's book, "The Theory of Matrices" and relies only on linear algebra. I don't know anything about the Dedekind-Weber result cited by Georges Elencwajg. I Vector bundles on complex projective spaces book that I found the book "Vector Bundles on Complex Projective Spaces" by Okonek et al to be very helpful. ISBN: OCLC Number: Description: viii, pages ; 23 cm. Series Title: Progress in mathematics (Boston. Serre and Swan showed that the category of vector bundles over a compact space X is equivalent to the category of finitely generated projective modules over the ring of continuous functions on X. Let X be an algebraic complex projective surface equipped with the euclidean topology and E a rank 2 topological vector bundle on X. It is a classical theorem of Wu ([Wu]) that E is uniquely. ] VECTOR BUNDLES AND PROJECTIVE MODULES module. Clearly T is an additive functor from the category of JC-vector bundles over X to the category of C(X)-modules. If £ is the trivial bundle E(Ç) = X x K", then T(^) is obviously a free C(X)-module on n generators. This book serves as an introduction to the topical question of classification of holomorphic Vector bundles on complex projective spaces. It includes many. Holomorphic vector bundles and the geometry of?n.- Stability and moduli spaces. (source: Nielsen Book Data) Summary These lecture notes are intended as an introduction to the methods of classi?cation of holomorphic vector bundles over projective algebraic manifolds X. To be as concrete as possible we have mostly restricted ourselves to the. Vector Bundles on Complex Projective Spaces | These lecture notes are intended as an introduction to the methods of classi?cation of holomorphic vector bundles over projective algebraic manifolds X. To be as concrete as possible we have mostly restricted ourselves to the case X = P. I'm following the book of Okonek, Schneider and Spindler, Vector Bundles on Complex Projective Spaces, and they say that is a useful exercise try to prove Bott's Formula that calculates the cohomology of exterior powers of the cotangent bundle on a projective space, by using induction.$\begingroup$It is a complex line, a$1$-dimensional complex vector space.$\endgroup$– Mariano Suárez-Álvarez Mar 8 '18 at 1$\begingroup$I think your fail is in the projective space definiton. Holomorphic Vector Bundles over Compact Complex Surfaces Vasile Brînzănescu (auth.) The purpose of this book is to present the available (sometimes only partial) solutions to the two fundamental problems: the existence problem and the classification problem for holomorphic structures in a given topological vector bundle over a compact complex. Vector bundles in Algebraic Geometry Enrique Arrondo Notes(*) prepared for the First Summer School on Complex Geometry (Villarrica, Chile December ) 1. The notion of vector bundle bers of the vector bundle have to be regarded as vector Size: KB. Existence of universal bundles: the Milnor join construction and the simplicial classifying space 63 The join construction 63 Simplicial spaces and classifying spaces 66 5. Some Applications 72 Line bundles over projective spaces 73 Structures on bundles and homotopy liftings 74 Embedded bundles and K-theory 77 File Size: KB. Among the bundles appearing in our classification one can find the Sasakura rank 3 vector bundle on the projective 4-space (conveniently twisted). The authors also propose a conjecture concerning the classification of globally generated vector bundles with $$c_1 \leq n. Vector Bundles on Complex Projective Spaces 作者: Spindler, Heinz 页数: 定价: 丛书: Modern Birkhäuser Classics ISBN: A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called s are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any operations of vector addition and scalar multiplication. F-projective space. Two F-vector bundles V and W over a finite complex B are said to be stably equivalent if the Whitney sums V EB fa and W EB h for some trivial F-vector bundles fa and h are isomorphic as F-vector bundles. The purpose of this paper is to study Schwarzenberger's property for. With the 2-sphere identified with the complex projective space (sub-bundles over paracompact spaces are direct summands) Let. Allen Hatcher, chapter 1 of Vector bundles and K-Theory, (partly finished book) web. Last revised on Aug at S. Ren: Order of the canonical vector bundle over configuration spaces of disjoint unions of spheres, arXiv/abs/ P. Salvatore: Configuration spaces on the sphere and higher loop spaces, Math. (), –Author: Shiquan Ren. which are assigned naturally to the base spaces of vector bundles and which encode information of such bundles. Beginning in the rst chapter with a summary of the basics of vector bundles, the project then turns in chapter 2 to the classi cation of vector bundles on a paracompact space, a necessary result to de ne the notion of a characteristic File Size: KB. The plan is for this to be a fairly short book focusing on topological K-theory and containing also the necessary background material on vector bundles and characteristic classes. Here is a provisional Table of Contents. At present only about half of the book is in good enough shape to be posted online, approximately pages. We can also define spaces in other ways, and then try to find cell complex structures for them. For example, the real projective n-space \({\mathbb{R}\textrm{P}^{n}}$$ is defined as the space of all lines through the origin in $${\mathbb{R}^{n+1}}$$. Each such line is determined by a unit vector, except that the negative of every vector is identified with the same line, so we can consider. Things become simpler if one passes from real vector spaces to complex vector spaces. The complex version of KO(X)g, called K(X)e, is constructed in the same way as KO(X)g but using vector bundles whose ﬁbers are vector spaces over Crather than R. The complex form of Bott Periodicity asserts simply that K(Se n)is Zfor nevenFile Size: 1MB. Let F be either the real number field R or the complex number field C and RP n the real projective space of dimension n. Theorems A and C in Hemmi and Kobayashi () [2] give necessary and sufficient conditions for a given F -vector bundle over RP n to be stably extendible to RP m for every m ⩾ : Yutaka Hemmi, Teiichi Kobayashi. So the only real projective spaces which can possibly be parallelisable are$\mathbb{RP}^1$,$\mathbb{RP}^3, \mathbb{RP}^7, \mathbb{RP}^{15}, \mathbb{RP}^{31}, \dots\$ We still need to determine which of these are actually parallelisable (the condition on the total Stiefel-Whitney class is a necessary condition, but not sufficient as the case of.

A classification of globally generated vector bundles E of rank two on P n with c 1 (E) = 2 was given in [1, Proposition ] obtaining, as a corollary, that E splits as a sum of two line bundles if n > 2 unless E is a twisted null-correlation bundle on P by: If the structure group of a vector bundle is reducible to $${GL(n,\mathbb{K})^{e}}$$, then it is called an orientable bundle; all complex vector bundles are orientable, so orientability usually refers to real vector tangent bundle of $${M}$$ (formally defined in an upcoming section) is then orientable iff $${M}$$ is a pseudo-Riemannian manifold $${M}$$, the structure.

of smooth complex projective varieties of dimension ≥ 4, over which every extension of line bundles splits. Introduction In algebraic geometry there is a rich history of studying when a vector bundle over a projective space splits, i.e.

is isomorphic to a direct sum of line bundles. Grothendieck. Algebraic Topology by Cornell University. This note covers the following topics: moduli space of flat symplectic surface bundles, Cohomology of the Classifying Spaces of Projective Unitary Groups, covering type of a space, A May-type spectral sequence for higher topological Hochschild homology, topological Hochschild homology of the K(1)-local sphere, Quasi-Elliptic Cohomology and its Power.

ALGEBRAIC TOPOLOGY: MATH BR NOTES 7 commutes, where ˚is a homeomorphism and ˚j Ex: Ex!Fx is a linear isomorphism. Tautological bundles on projective spaces and Grassmannians.

So far we’ve only looked at real vector bundles, but we will now consider complex ones. Deﬁnition Let CPn, or Pn C denote x: xˆCn+1: xis a 1 File Size: KB.Geometry and stability of tautological bundles on Hilbert schemes of points Stapleton, David, Algebra & Number Theory, ; Quiver flag varieties and multigraded linear series Craw, Alastair, Duke Mathematical Journal, ; Dimension estimates for Hilbert schemes and effective base point freeness on moduli spaces of vector bundles on curves Popa, Mihnea, Duke Mathematical Journal,   In developing the tools necessary for the study of complex manifolds, this comprehensive, well-organized treatment presents in its opening chapters a detailed survey of recent progress in four areas: geometry (manifolds with vector bundles), algebraic topology, differential geometry, and partial differential equations.

Subsequent chapters then develop such topics as Hermitian exterior algebra.

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