Grothendieck that algebraic vector bundles on the projective line are direct sums of line bundles. The object of this paper is to study continuous vector bundles, over real algebraic varieties, admitting an algebraic structure. In particular, kumar and murthy established existence of algebraic vector bundles of rank 63 with. This is the definition taken in positivity in algebraic geometry ii, by robert lazarsfeld, and in example 6. Analytic cycles and vector bundles on noncompact algebraic varieties. Vector bundles on algebraic varieties school of mathematics, tifr. The aim of the colloquium was to highlight recent developments in the general area of vector bundles as well as principal bundles of both affine and projective varieties. In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. Mumford was awarded the fields medal in 1974 for his work on moduli spaces and algebraic surfaces. Pdf complex vector bundles on real algebraic varieties of. Almost homogeneous algebraic varieties under torus action. Vector bundles of a new type, called strati ed algebraic vector bundles, are introduced and investigated in this paper. For algebraic geometry, we as sume familiarity with basic concepts like algebraic varieties, alge braic curves, vector bundles. Monads appeared in a wide variety of contexts within algebraic geometry, and they are very useful when we want to construct vector bundles with.
On the study of algebraic varieties via vector bundles. Here we mention only that the results were suggested by those of harris 1984, 1985, which in turn were suggested. The study of algebraic vector bundles of higher ranks began in 1957, when it was shown by a. Outline 1 vector bundles topological vector bundles algebraic vector bundles 2 a1homotopy theory motivation the category objectwise homotopy theory local homotopy theory a1homotopy theory on morels classifying theorem applications 3 stable homotopy theory over a. Throughout this paper, the base field is the field c of complex numbers. Let e be an algebraic vector bundle of rank n over c2 with an action of. Almost homogeneous algebraic varieties under torus action, manifolds tokyo 1973, 373 381, proceedings of the international conference on manifolds and related. About vector bundles on algebraic varieties mathematics. Some applications of vector bundles in algebraic geometry.
Assume that dim x 1, then, for each closed field integer r dim x, there is a simple, a fortiori, indecomposable r on vector bundle of rank x horeover, 118 if dim x 2, there are algebraic vector bundles k varieties t which parametrize bijectively families of simple vector r on x and sup dim t r t bundles of rank 00. An amodule p is projective if it is a direct summand of a free module. Introduction let x be a smooth projective variety over. G valuative criteria for families of vector bundles on algebraic varieties. Cycles and bundles are intrinsic invariants of algebraic varieties close connections going back to grothendieck work with quasiprojective varieties over a. On adjoint bundles of ample vector bundles, complex algebraic varieties, proc. Desigularization of moduli varieties of vector bundles on curves.
Introduction let g be a reductive complex affine algebraic group and z a complex affine gvariety with a gfixed base point z 0 ez. Schemes are algebraic schemes over an algebraically closed field k c for clarity. Vector bundles of a new type, called strati edalgebraic vector bundles, are introduced and investigated in this paper. The study of vector bundles over algebraic varieties has been stimulated over the last few years by successive waves of migrant concepts, largely from mathematical physics, whilst retaining its roots in old questions concerning subvarieties of projective space. We investigate vector bundles on real algebraic varieties. The bayreuth meeting on complex algebraic varieties focussed on the classification of algebraic varieties and topics such as vector bundles, hodge theory and hermitian differential geometry. The modern definition of a vector bundle which works for manifolds, analytic spaces and schemes at once, in fact arbitrary ringed spaces is that of a locally free sheaf of modules. Lectures on moduli spaces of vector bundles on algebraic varieties. Proceedings of the international symposium on algebraic. To prove theorem ii, one constructs an algebraic family of. Locally free sheaves come equipped with the standard module operations, but these give back locally free sheaves. The purpose of the colloquium was to highlight recent developments in the general area of vector bundles as well as principal bundles on both a. In no case do i claim it is a survey on moduli spaces of vector bundles on algebraic projective varieties.
Assume that dim x 1, then, for each closed field integer r dim x, there is a simple, a fortiori, indecomposable r on vector bundle of rank x horeover, 118 if dim x 2, there are algebraic vector bundles kvarieties t which parametrize bijectively families of simple vector r on x and sup dim t r t bundles of rank 00. Papers presented at the bombay colloquium 1984 tata institute of fundamental research, bombay studies in mathematics on free shipping on qualified orders. The next topic that we will treat is a particular class of vector bundles or sheaves, namely arithmetically cohenmacaulay acm and ulrich sheaves, tra ditionally important in commutative algebra and representation theory of rings. Function theory of finite order on algebraic varieties. This vector bundle turns out to be a sufficiently good invariant of p so that the relation obtained in this way between vector bundles on x x 82 and vector bundles on x gives. In terms of algebraic geometry, they are defined as stable vector bundles e of rank 2 on the complex projective space 3, having c1e 0 and with the prescribed. As hartshorne has noted, it is helpful to regard vector bundles as a special kind of reflexive sheaves.
We obtain a partial solution of this problem and present two applications. Andrey tyurin algebraic geometry, written during his whole. Vector bundles and brillnoether theory, in current topics in complex algebraic geometry, cambridge univ. It also develops the theory of vectorbundles over an af. Vector bundles usc dana and david dornsife college of. On a family of algebraic vector bundles, number theory, algebraic geometry and commutative algebra, in honor of akizuki, y. Automorphic vector bundles on connected shimura varieties 93 a discussion of the relation of these results to those of other authors will be given in the sequel to this paper.
On equivariant vector bundles on an almost homogeneous variety volume 57 tamafumi kaneyama. Algebraic geometry lecture notes mit opencourseware. Confusion on definition of vector bundles in algebraic geometry. In real algebraic geometry, the role of algebraic, semi algebraic and nash vector bundles is rmly established. Algebraic vector bundles are much more complicated than topological vector bundles. We want to introduce the language of vector bundles as a method to describe.
Algebraic vector bundles over real algebraic varieties. Algebraic varieties we always assume working over some algebraically closed eld k. Equivariant algebraic vector bundles over a product of affine varieties kayo masuda received april 7, 1996 0. Algebraic geometry seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly. Any compact real algebraic variety xadmits a nonsingular strati cation ssuch that each strati ed algebraic f vector bundle on xis isomorphic in the category. Appendix d on page 501 develops basic concepts of homological algebra and applies this to sheaves and rings.
Pdf complex vector bundles on real algebraic varieties. On equivariant vector bundles on an almost homogeneous. Thus, we turn our attention to oriented vector bundles of rank n 1, which we refer to as the critical rank. It is an additional structure which belongs to the vector bundle. Browse other questions tagged algebraicgeometry vectorbundles coherent. Some applications of vector bundles in algebraic geometry daniele. As motivation, we discuss vector bundles on real manifolds. In real algebraic geometry, the role of algebraic, semialgebraic and nash vector bundles is rmly established. Vector bundles in algebraic geometry enrique arrondo notes prepared for the first summer school on complex geometry villarrica, chile 79 december 2010 1.
But kis at most counxtably dimensional as a vector space over k, so t a. By appointment, in 380383m third floor of the math building. Algebraic geometry i complex projective varieties david. Analytic cycles and vector bundles on noncompact algebraic varieties 3 in the case of line bundles, we are able to give a new proof of grauerts theorem based on the lzmethods for the 0operator and a certain lineari zation trick w167 19 and 20, a proof which does give some growth conditions. Complex algebraic varieties proceedings of a conference. The study of line bundles on projective varieties is a classical problem in algebraic geometry cf. In this paper, all varieties are defined over an algebraically closed. And it also makes clear what is part of the structure the sheaf of modules and what is a property local triviality.
Ci,c2 of rank two, lstable vector bundles e, on pbundles of. Analytic cycles and vector bundles on noncompact algebraic. Introduction let x be a smooth projective variety over the complex number field. Mumford worked in the fields of algebraic gemetry in the 60s and 70s, concentrating especially on the theory of moduli spaces. Most of the articles in this volume are closely related to talks given at the conference. Xthere is an algebraic vector bundle of rank two with those chern classes. Algebraic surfaces and holomorphic vector bundles universitext kindle edition by robert friedman.
A coherent sheaf t on an algebraic variety x is called reflexive if the natural map of t to its double dual t is an isomorphism. Strati ed algebraic vector bundles on a compact real algebraic variety are in some sense stable with respect to strati cations. Ci,c2 of rank two, lstable vector bundles e, on p bundles of arbitrary dimension, with fixed chern classes. For large classes of real varieties, we obtain explicit information.
Algebraic transformation group and algebraic varieties, pp. A function theory of finite order on algebraic varieties. Many people have made important contributions without even being mentioned. Vector bundle, algebraic encyclopedia of mathematics. Actually this definition simplifies lots of constructions for vector bundles. This vector bundle turns out to be a sufficiently good invariant of p so that the relation obtained in this way between vector bundles on x x 82 and vector bundles on x gives the periodicity theorem. It has now been four decades since david mumford wrote that algebraic geometry seems to have acquired the reputation of being esoteric, exclusive, and. Strati edalgebraic vector bundles form an intermediate category between algebraic and semialgebraic vector bundles. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information coherent sheaves can be seen as a generalization of vector bundles. Similarly, in dimension three, a positive answer to question1follows from the work of kumar and murthy km82, theorem 2.
Our goal is to construct rank 2 real and complex strati. Representability in the a1homotopy category implies x 7. Some applications of vector bundles in algebraic geometry daniele faenzi email address. On the periodicity theorem for complex vector bundles. The aim of these notes is to present a general algebraic setting based on the euler isomorphism for complexes of vector spaces as in the book by gelfand, kapranov, and zelevinsky, and on some self. S desigularization of moduli varieties of vector bundles on curves. Appendix c on page 489 develops the topological concept of vector bundles. Introduction to intersection theory in algebraic geometry lectures. This enables us to get rid of the smoothness condition and to work in the analytic category.
Let x be an algebraic variety and a ox the algebra of. Use features like bookmarks, note taking and highlighting while reading algebraic surfaces and holomorphic vector bundles universitext. Examples to keep in mind are the tangent bundle to a manifold, and the mobius strip. In chapter 5 we deal with moduli spaces msub x,l 2. I am confused about the connection between vector bundles and locally free sheaves over schemes. The sheaftheoretic interpretation of vector bundles has the advantage that vector bundles on a locally noetherian scheme are included in the abelian category of coherent sheaves. Geometric realization of tshaped root systems and counterexamples to hilberts fourteenth problem, 2002. Algebraic stacks and moduli of vector bundles impa. On the study of algebraic varieties via vector bundles tadakazu sawada mathematical institute, tohoku university, d1 introduction the basic object of study in algebraic geometry is the locus of roots of a. 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. Automorphic vector bundles on connected shimura varieties. A vector bundle of rank r or line bundle if r 1 over an algebraic variety.
Strati ed algebraic vector bundles form an intermediate category between algebraic and semi algebraic vector bundles. Proceedings of the international symposium on algebraic geometry, kyoto, 1977 15. Confusion on definition of vector bundles in algebraic. Moduli of vector bundles on curves with parabolic structures. Ci, c2 of rank r, hstable vector bundles on either minimal rational surfaces or on algebraic k3 surfaces. Oriented algebraic vector bundles of rank non q 2n 1 are easy to describe, essentially because the classi. Vector bundles on surfaces and higherdimensional varieties, and their moduli see also 14d20. Download it once and read it on your kindle device, pc, phones or tablets. An internationalcolloquiumon vector bundles on algebraic varieties was held at the tata institute of fundamental research, bombay from january 9 to january 16, 1984.
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