Mirror symmetry for concavex vector bundles on projective spaces mirror symmetry for concavex vector bundles on projective spaces. 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. Algebraic cohomology of the moduli space of rank 2 vector. Hartshorne algebraic vector bundles on projective spaces. Some applications of vector bundles in algebraic geometry. Introduction in algebraic geometry there is a rich history of studying when a vector bundle over a projective space splits, i. Mirror symmetry for concavex vector bundles on projective. Boundaries of holomorphic chains within vector bundles over complex projective space ronald a. For full access to this pdf, sign in to an existing account, or purchase an annual subscription. A vector bundle is a family of vector spaces that is locally trivial, i. Walker boundaries of holomorphic chains within vector bundles. Pdf vector bundles on projective 3space researchgate. A vector bundle for which a trivialization exists is called trivializable accordingly, the local triviality condition in the definition of topological vector bundles def.
Lecture notes geometry of manifolds mathematics mit. Nontrivial hilbertspace vector bundles over classical phase space provide one such framework. 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. This construction was presented by the author in the germanspanish. Nef vector bundles on a projective space with first chern class three. 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. 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. 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. 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. 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. I recall that i found the book vector bundles on complex projective spaces by okonek et al to be very helpful. Rohatgi the davidson academy of nevada october 31, 2016. Here bundle simply means a local product with the indicated. 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.
We shall consider 2dimensional complex vector bundles over the 3dimensional complex projective space p3. Line bundles on projective spaces preliminary draft 3 our description given here. 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. 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. Using linear algebra to classify vector bundles over p1. In respect of moduli space, vector bundles a k3 surface look like picard varieties in the preceding section. In it the reader will find results from the literature, as well as new contributions to the field. 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. We shall be concerned with vector bundles over x, i. Here we give some examples in the case of mathematical instanton bundles and their moduli spaces. Vector bundles on projective space takumi murayama december 1, 20 1 preliminaries on vector bundles let xbe a quasiprojective variety over k.
Q spanned by the fundamental classes of algebraic cycles, in terms of the algebraic cohomology ring of the jacobian j c. Faltings construction of the moduli space of vector bundles on a smooth projective curve georg hein abstract. In this thesis we discuss the theory of vector bundles with real structure on the projective line. This extends classical work by grothendieck classifying complex vector bundles on the projective line. Intuitively, one should think about e as the collection exx2m of vector spaces of rank r, smoothly parametrized by x 2 m. Extendibility of negative vector bundles over the complex. Notes by gergely berczi, michael groechenig and geordie williamson. Every continuous 2dimensional vector bundle over p3. 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.
This paper identifies the algebraic cohomology ring h a. By schwarzenbergers property, a complex vector bundle of dimension t. Faltings construction of the moduli space of vector. Frobenius morphism and vector bundles on cycles of projective lines igor burban abstract. Vector bundles on complex projective spaces springerlink. A family of vector spaces over xis a morphism of varieties e.
This bundle, denoted by t pn logd, was originally introduced by deligne and saito cf. A vector bundle on x is an algebraic ber space in vector spaces or, equivalently, a locally free coherent o xmodule. 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. 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. Shows that the moduli space of stable rank 2 vector bundles on. Every continuous 2dimensional vector bundle over p3 admits a. Short supplementary sections describe more advanced. Moduli of rank2 vector bundles, theta divisors, and the geometry of curves in projective space. Then associated with momentangle manifolds, we further study. This gives an answer on a question of paul monsky, which appeared in his study of the hilbertkunz theory for plane cubic curves. The topology of fiber bundles lecture notes ralph l. In this paper we describe the action of the frobenius morphism on the indecomposable vector bundles on cycles of projective lines.
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. The difference between a vector space and the associated af. An algebraic rank2 vector bundle o on the projective plane pip2ii is. Anticanonical divisors of a moduli space of parabolic vector bundles of half weight on. Pdf topological classification of complex vector bundles. Pdf projective bundles over small covers and topological. Multiple quantum products in toric varieties multiple quantum. Find materials for this course in the pages linked along the left. We will prove these equivalences, which are a model for the subsequent discussion of the brauer group in algebraic geometry.
The only two vector bundles with base space a circle and onedimensional. Representations of elementary abelian pgroups and vector. Just as ktheory classi es vector bundles on a space, the. Vector bundles on projective space university of michigan. Vector bundles and connections universiteit utrecht. 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. 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. However, for the purposes of this paper, our terminology should be. We have attempted to give an informal presentation of the main results, addressed to a general audience. Among the bundles appearing in our classification one can find the sasakura rank 3 vector bundle on the projective 4space conveniently twisted. Boundaries of holomorphic chains within vector bundles. 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.
Notes on principal bundles and classifying spaces stephen a. Barthmoduli of vector bundles on the projective plane. We first give a necessary and sufficient condition for such a projectivization to be a small cover. A connection between vector bundles over smooth projective curves and representations of quivers dhruv w. 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. Kempf for a vector bundle on projective space to be. When the genus of x is 0, vector bundles on x can be classi ed birkho, grothendieck. One also says that the data consisting of an open cover u i. Algebraic vector bundles on projective space springerlink.
They allow for the possibility of having different, nonequivalent quantisations, as opposed to the uniqueness of the trivial bundle. Notes of talks by tamas hausel in oxford, trinity term, 2010. X, there exists a neighborhood u x such that there is an. 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. On extending vector bundles over projective space, the quarterly journal of mathematics, volume 17, issue 1, 1 january 1966, pages 1418, s. Is the category of vector bundles over a topological space. Vector bundles on projective space takumi murayama december 1, 20 1 preliminaries on vector bundles let xbe a quasi projective variety over k. A connection between vector bundles over smooth projective. Indecomposable vector bundles and stable higgs bundles. No distinction will be made between holomorphic vector bundles and locally. Walker penn state harrisburg march 19, 2011 ronald a. I dont know anything about the dedekindweber result cited by georges elencwajg. This is part of the more general problem to find vector bundles of small rank on large projective spaces. It is known that there are indecomposable bundles of rank.