The group kog pn of equivalence classes of stable vector bundles over real projective space is a. 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. Among the bundles appearing in our classification one can find the sasakura rank 3 vector bundle on the projective 4space conveniently twisted. 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. Algebraic vector bundles on projective space springerlink.
Vector bundles on projective space takumi murayama december 1, 20 1 preliminaries on vector bundles let xbe a quasi projective variety over k. Barthmoduli of vector bundles on the projective plane. Find materials for this course in the pages linked along the left. Here bundle simply means a local product with the indicated.
It is known that there are indecomposable bundles of rank. Using linear algebra to classify vector bundles over p1. Boundaries of holomorphic chains within vector bundles over complex projective space ronald a. They allow for the possibility of having different, nonequivalent quantisations, as opposed to the uniqueness of the trivial bundle. The topology of fiber bundles lecture notes ralph l. X, there exists a neighborhood u x such that there is an. We shall be concerned with vector bundles over x, i. Shows that the moduli space of stable rank 2 vector bundles on.
Anticanonical divisors of a moduli space of parabolic vector bundles of half weight on. This bundle, denoted by t pn logd, was originally introduced by deligne and saito cf. 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. Boundaries of holomorphic chains within vector bundles. Walker boundaries of holomorphic chains within vector bundles. 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. We will prove these equivalences, which are a model for the subsequent discussion of the brauer group in algebraic geometry. Pdf vector bundles on projective 3space researchgate. Short supplementary sections describe more advanced.
This extends classical work by grothendieck classifying complex vector bundles on the projective line. Some applications of vector bundles in algebraic geometry. However, for the purposes of this paper, our terminology should be. A problem list 119 in particular one can ask what are the irreducible components of m, what are their dimensions, is there a universal family of bundles on m in which case m would be a fine moduli space. I recall that i found the book vector bundles on complex projective spaces by okonek et al to be very helpful. Vector bundle over projective space mathematics stack. Let n c be the moduli space of stable holomorphic vector bundles of rank 2 and fixed determinant of odd degree, over a smooth projective curve c. Rohatgi the davidson academy of nevada october 31, 2016. Vector bundles over an elliptic curve 415 embedded biregularly in some projective space. Representations of elementary abelian pgroups and vector. This construction was presented by the author in the germanspanish. Notes by gergely berczi, michael groechenig and geordie williamson. In this paper, complex vector bundles of rank r over 8dimensional spinc manifolds are classified in terms of the chern classes of the complex vector bundles and the cohomology ring of the manifolds, where r 3 or 4. On extending vector bundles over projective space, the quarterly journal of mathematics, volume 17, issue 1, 1 january 1966, pages 1418, s.
Hartshorne algebraic vector bundles on projective spaces. Q spanned by the fundamental classes of algebraic cycles, in terms of the algebraic cohomology ring of the jacobian j c. One also says that the data consisting of an open cover u i. Line bundles on projective spaces preliminary draft 3 our description given here. Indecomposable vector bundles and stable higgs bundles. In it the reader will find results from the literature, as well as new contributions to the field. A vector bundle is a family of vector spaces that is locally trivial, i. Pdf topological classification of complex vector bundles. Walker penn state harrisburg march 19, 2011 ronald a. Intuitively, one should think about e as the collection exx2m of vector spaces of rank r, smoothly parametrized by x 2 m.
Vector bundles on projective space takumi murayama december 1, 20 1 preliminaries on vector bundles let xbe a quasiprojective variety over k. We shall consider 2dimensional complex vector bundles over the 3dimensional complex projective space p3. In 1993 faltings gave a construction of the moduli space of semistable vector bundles on a smooth projective curve x over an algebraically closed eld k. The only two vector bundles with base space a circle and onedimensional. Every continuous 2dimensional vector bundle over p3. This is part of the more general problem to find vector bundles of small rank on large projective spaces. Here we give some examples in the case of mathematical instanton bundles and their moduli spaces. Just as ktheory classi es vector bundles on a space, the. The difference between a vector space and the associated af. The statement and proof of the kronecker pencil lemma can be found in gantmachers book, the theory of matrices and relies only on linear algebra. Im using the letter m to denote the base space of the vector bundle as a concession to the fact that in most of the applications well be.
A vector bundle on x is an algebraic ber space in vector spaces or, equivalently, a locally free coherent o xmodule. Then associated with momentangle manifolds, we further study. We first give a necessary and sufficient condition for such a projectivization to be a small cover. 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. Extendibility of negative vector bundles over the complex projective space dedicated to the memory of professor masahiro sugawara mitsunori imaoka received march 24, 2005 revised july 4, 2005 abstracf. When the genus of x is 0, vector bundles on x can be classi ed birkho, grothendieck. Vector bundles in this section we recall some basics on vector bundles. 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. Nef vector bundles on a projective space with first chern class three. Extendibility of negative vector bundles over the complex. This gives an answer on a question of paul monsky, which appeared in his study of the hilbertkunz theory for plane cubic curves. 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.
By schwarzenbergers property, a complex vector bundle of dimension t. I dont know anything about the dedekindweber result cited by georges elencwajg. In this paper we describe the action of the frobenius morphism on the indecomposable vector bundles on cycles of projective lines. Introduction in algebraic geometry there is a rich history of studying when a vector bundle over a projective space splits, i. Notes on principal bundles and classifying spaces stephen a. Frobenius morphism and vector bundles on cycles of projective lines igor burban abstract. 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. Is the category of vector bundles over a topological space. Mirror symmetry for concavex vector bundles on projective spaces mirror symmetry for concavex vector bundles on projective spaces. 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. Every continuous 2dimensional vector bundle over p3 admits a. Vector bundles on projective space university of michigan. You shouldnt expect vector bundles to form an abelian category because they are projective module objects over local ring objects in a category of sheaves over a space.
Nontrivial hilbertspace vector bundles over classical phase space provide one such framework. Moduli of rank2 vector bundles, theta divisors, and the geometry of curves in projective space. Notes of talks by tamas hausel in oxford, trinity term, 2010. In respect of moduli space, vector bundles a k3 surface look like picard varieties in the preceding section. Faltings construction of the moduli space of vector bundles on a smooth projective curve georg hein abstract. In other words, there is really no more reason to expect this of vector bundles than you would of a category of finitely generated projective modules over a ring. We have attempted to give an informal presentation of the main results, addressed to a general audience. Mitchell august 2001 1 introduction consider a real nplane bundle.
Vector bundles and connections universiteit utrecht. A connection between vector bundles over smooth projective. Vector bundles on complex projective spaces springerlink. Kempf for a vector bundle on projective space to be. For full access to this pdf, sign in to an existing account, or purchase an annual subscription. 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. A vector bundle for which a trivialization exists is called trivializable accordingly, the local triviality condition in the definition of topological vector bundles def. An mdimensional vector bundle v over a space a is called extendible to a space bia when there exists an mdimensional vector bundle over b whose restriction to a is isomorphic to v.
Multiple quantum products in toric varieties multiple quantum. 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. Lecture notes geometry of manifolds mathematics mit. Algebraic cohomology of the moduli space of rank 2 vector. No distinction will be made between holomorphic vector bundles and locally. A family of vector spaces over xis a morphism of varieties e. Pdf projective bundles over small covers and topological.
Mirror symmetry for concavex vector bundles on projective. In this thesis we discuss the theory of vector bundles with real structure on the projective line. This paper identifies the algebraic cohomology ring h a. An algebraic rank2 vector bundle o on the projective plane pip2ii is. As an application, we got that two rank 3 complex vector bundles over 4dimensional complex projective spaces cp4 are isomorphic if and only if they have the same chern classes. Faltings construction of the moduli space of vector. Classically, schwarzenberger 11, 4, appendix i studied extendibility of vector bundles over the real or complex projective spaces. This paper investigates the projectivization of real vector bundles over small covers.
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