Vector bundles over projective space pdf

Vector bundles on cones over projective schemes springerlink. Let a be an a ne space modelled on the fvector space v and let b. Much is known in this direction, for example the topological and holomorphic kgroups of all complex projective spaces are isomorphic. Some topics on higgs bundles over projective varieties and. Quillen, d higher algebraicktheory, algebraicktheory i. Walker boundaries of holomorphic chains within vector bundles. Clearly t is an additive functor from the category of jc vector bundles over x to the category of cxmodules. A canonical treatment of line bundles over general projective spaces 3 2. For some time, it has been assumed that a similar correspondence exists between topological vector bundles over a compact haus dorff space x and finitely generated projective modules over the ring of con. Rohatgi mentored by vishal arul a connection between vector bundles over smooth projective curves and representations of quivers. Let be the canonical line bundle over the complex projective space cp, and for an integer iii.

Absolutely split real algebraic vector bundles over a real. Part ii is devoted to the classification of vector bundles over an elliptic curve. Results concerning rationality and irreducibility are known for moduli spaces of stable bundles over curves, but as far as i know not yet for surfaces. The purpose of this paper is to study schwarzenbergers property for vector bundles over the quaternionic projective space hpn. Workshop on moduli spaces of vector bundles at the university of essen in.

Thanks for contributing an answer to mathematics stack exchange. Yoshida 8, 9 and so on, and that of vector bundles over the quaternionic projective spaces by 6, 7. Vector bundles on projective space university of michigan. Walker penn state harrisburg march 19, 2011 ronald a. We consider only complex vector bundles, and thus a kdimensional vector bundle means a ck vector bundle. Stable extendibility of vector bundles over real projective spaces and bounds for the schwarzenberger numbers bk teiichi kobayashi and toshio yoshida received april 8, 2003 revised july 7, 2003 abstract. The classification of holomorphic algebraic vector bundles on complex. In the rst two chapters, xis a smooth projective curve with a marked point p over an algebraically closed eld, and the vector bundles are endowed with parabolic structures, i. Vector bundles in this section we recall some basics on vector bundles.

The only two vector bundles with base space a circle and onedimensional. Boundaries of holomorphic chains within vector bundles. Vector bundles on projective space takumi murayama december 1, 20 1 preliminaries on vector bundles let xbe a quasi projective variety over k. Recall here that, for a ring r,anrmodule e is said to be. Vector bundles over an elliptic curve 415 embedded biregularly in some projective space.

The group kog pn of equivalence classes of stable vector bundles over real projective space is a. Short supplementary sections describe more advanced topics, further results, and unsolved problems. Indecomposable vector bundles and stable higgs bundles over smooth projective curves by olivier schiffmann abstract we prove that the number of geometrically indecomposable vector bundles of xed rank rand degree dover a smooth projective curve xde ned over a nite eld is given by a polynomial depending only on r. A connection between vector bundles over smooth projective curves and representations of quivers dhruv w. This space is called the moduli space of semistable vector bundles and we denote it. Extendibility of vector bundles over the real projective spaces and the standard lens spaces are studied extensively by kobayashimakiyoshida 8, 9 and so on, and that of vector bundles over the quaternionic projective spaces by 6, 7. It is known that there are indecomposable bundles of rank. This bundle, denoted by t pn logd, was originally introduced by deligne and saito cf. Vector bundle over projective space mathematics stack. In it the reader will find results from the literature, as well as new contributions to the field. Indecomposable vector bundles and stable higgs bundles. We consider only complex vector bundles, and thus a kdimensional vector bundle means a ck. We start with recalling the general technique used in a study of vector bundles on singular projective curves, see 3, 1, 2. 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.

In summary, this book provides a thorough introduction to the theory of the correspondence between modular representations of elementary abelian groups and vector bundles over projective space. Rohatgi mentored by vishal arul 6th annual mitprimes conference may 21, 2016 dhruv w. On extending vector bundles over projective space, the quarterly journal of mathematics, volume 17, issue 1, 1 january 1966, pages 1418, s. A connection between vector bundles over smooth projective. It is shown that all line bundles are tensor powers of the hopf bundle and all vector bundles are. Vector bundles on complex projective spaces with an. Algebraic vector bundles on projective space 299 w 2.

The groups k o we now assume that a is a regular noetherian ring. Vector bundles and connections universiteit utrecht. Vector bundles on projective spaces department of mathematics. Some applications of computer algebra to vector bundles on projective spaces. We shall be concerned with vector bundles over x, i. Hartshorne algebraic vector bundles on projective spaces. A family of vector spaces over xis a morphism of varieties e. Vector bundles over the projective space pn k are one of the main subjects in both algebraic geometry and commutative algebra. A canonical treatment of line bundles over general. For some time, it has been assumed that a similar correspondence exists between topological vector bundles over a compact hausdorff space x and finitely generated projective modules over the ring of con. We have attempted to give an informal presentation of the main results, addressed to a general audience.

If k is the complex field then it has been shown by serre 9 that the algebraic and. Pdf topological classification of complex vector bundles. This text deals with vector bundles over a curve x. A moduli space of stable vector bundles on a smooth, algebraic variety x is a scheme whose points are in natural bijection to isomorphic classes of stable vector bundles on x. 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. An algebraic variety called a variety for short is any a ne, quasia ne, projective, or quasiprojective variety. 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. Faltings construction of the moduli space of vector bundles on a. Representations of elementary abelian pgroups and vector. Dear arrow, you are right in some sense but did you read footnote 1 to definition 26. Extendibility of negative vector bundles over the complex. On vector bundles over algebraic and arithmetic curves. Kempf for a vector bundle on projective space to be.

Over the field of one element, vector bundles over ndimensional projective spaces are considered. Let c be a smooth projective connected algebraic curve over some eld k. Stably extendible vector bundles over the quaternionic. Over a riemann surface, the hermitianyangmills equations for a connection simply say that the connection is. 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. Much is known in this direction, for example the topological and holomorphic k groups of all complex projective spaces are isomorphic. Schwarzenberger sc, hi has shown the fact that a kdimensional fvector bundle v over fpn. The most fundamental result in this area is the theorem due to grothendieck which asserts that any holomorphic vector bundle over p1 k splits into a direct sum of line bundles. In section 3 we study stable extendibility of cvector bundles over rp n and prove theorem 5, theorem 6, theorem 7 and corollary 8. This is part of the more general problem to find vector bundles of small rank on large projective spaces. Vector bundles on projective space takumi murayama december 1, 20 1 preliminaries on vector bundles let xbe a quasiprojective variety over k.

Some applications of vector bundles in algebraic geometry. Our goal is to describe the action of the frobenius morphism on the set of indecomposable vector bundles on e. A real or complex vector bundle on xor over x is a topological space ewith a continuous map e. In this paper, we study the order of the canonical vector bundle over configuration spaces of projective spaces as well as of the cartesian products of a projective space and a euclidean space. Vector bundles on complex projective spaces springerlink. Boundaries of holomorphic chains within vector bundles over complex projective space ronald a. Stable extendibility of some complex vector bundles over lens spaces and schwarzenbergers theorem hemmi, yutaka and kobayashi, teiichi, hiroshima mathematical journal, 2016 quasipositive curvature on homogeneous bundles tapp, kristopher, journal of differential geometry, 2003. Introduction in algebraic geometry there is a rich history of studying when a vector bundle over a projective space splits, i. Xand a real or complex vector space structure on each. Order of the canonical vector bundle over configuration spaces of projective spaces. Using linear algebra to classify vector bundles over p1.