Volume 220, number 4 physics letters b april 1989 connections on vector bundles over super riemann surfaces mark rakowski and george thompson international centre for theoretical physics, 4100 trieste, italy received 1 august 1988 we show that the globally inequivalent offshell n1 super yangmills theories in two dimensions classify the superholomorphic structures on vector bundles. Chapter iv, which presents analytic continuation and the construction of the riemann surface of an irreducible algebraic equation pz,w 0, represent lectures of raghavan narasimhan. The following are lecture notes prepared for a fourhours minicourse on higgs bundles i had the opportunity to give at heidelberg university in november and december 2014. Stable vector bundles on algebraic surfaces internet archive. Line integral from vector calculus over a closed curve. Lectures on vector bundles cambridge studies in advanced mathematics 54. I would also recommend griffithss introduction to algebraic curves a beautiful text based on lectures. E m a vector bundle on m, then a metric on e is a bundle map k. Algebraic surfaces and holomorphic vector bundles robert. So, we obtain a sheaf of oxmodules e oxe over x, locally isomorphic to or x. Get your kindle here, or download a free kindle reading app. Introduction to holomorphic vector bundles on compact riemann. A remark on universal coverings of holomorphic families of riemann surfaces imayoshi, yoichi and nishimura, minori, kodai. Buy lectures on vector bundles over riemann surfaces.
Download this book grew out of lectures on riemann surfaces given by otto forster at the universities of munich, regensburg, and munster. E of regular sections of e over u has a structure of module over the algebra oxu. It provides a concise modern introduction to this rewarding subject, as well as presenting methods used in the study of complex manifolds in the special case of complex dimension one. In mathematics, vector bundles on algebraic curves may be studied as holomorphic vector bundles on compact riemann surfaces. Concerning the vector bundle theory, his famous papers, with m. I have encountered two formulations for vector bundles and clearly there are many more, and i am trying to understand why it is that they are equivalent.
Narasimhan1 and by himself2, inaugurated the modern theory of holomorphic vector bundles on compact riemann surfaces. On schottky vector bundles over riemann surfaces by carlos armindo arango florentino doctor of philosophy in mathematics state university of new york at stony brook 1997 let xbe a compact riemann surface of genus g. Vector bundles on riemann surfaces and conformal field. Its aim is to give an introduction to this rich and beautiful subject, while presenting methods from the theory of complex manifolds which, in the special case of one complex variable, turn out to be particularly elementary and transparent. 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. Other readers will always be interested in your opinion of the books youve read. On vector bundles over algebraic and arithmetic curves. An introduction to the topology of the moduli space of stable. This content was uploaded by our users and we assume good faith they have the permission to share this book. Lectures on the moduli stack of vector bundles on a curve.
We study a natural map from representations of a free group of rank g in gln,c, to holomorphic vector bundles of degree 0 over a compact riemann surface x of genus g, associated with a schottky uniformization of x. This text deals with vector bundles over a curve x. Holomorphic bundles over elliptic manifolds sources. There is a class of metrics on the tangent bundle tm of a riemannian manifold m, g oriented, or nonoriented, respectively, which are naturally constructed from the base metric g 15. In chapter 4 the author proves the famous riemannroch theorem for the case of vector. This book deserves very serious consideration as a text for anyone contemplating giving a course on riemann surfaces. Quotient stacks let xbe a scheme say over some eld kin order to avoid a atness condition in what follows and gbe an algebraic group acting on x. We show that the globally inequivalent offshell n1 super yangmills theories in two dimensions classify the superholomorphic structures on vector bundles over super riemann surfaces. The goal of these lectures was to acquaint researchers in 4manifold topology with the classification of algebraic surfaces and with methods for. It also deals quite a bit with noncompact riemann surfaces, but does include standard material on abels theorem, the abeljacobi map, etc.
But there are many other interesting vector bundles. This book grew out of lectures on riemann surfaces which the author gave at the universities of munich, regensburg and munster. Lectures on riemann surfaces a very attractive addition to the list in the form of a wellconceived and handsomely produced textbook based on several years lecturing experience. However, since his not commutative, one has to be careful when writing down formulas.
Lectures on vector bundles on riemann surfaces gunning r. Algebraic moduli problems, moduli of vector bundles for. Full text of schottky uniformization and vector bundles. Lectures given in the minischool on moduli spaces at the banach center warsaw 2630 april 2005. Differential geometry of holomorphic vector bundles on a curve. 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. Our aim is however to study the poisson structure on the moduli space of locally flat vector bundles on a riemann surface with holes i. Differential geometry of complex vector bundles shoshichi. On the other hand, the classification of vector bundles on projective curves is closely related to the study of cohenmacaulay modules on surface singularities, due to the work of kahn 10.
Complex vector bundles over compact riemann surfaces. Construct a ranktwo vector bundle over the smooth quadric x. Vector bundles on riemann surfaces and conformal field theory arnaud beauville introduction the main character of these lectures is a. Quantum cohomology of the moduli space of stable bundles over a. Ariemannsurfaceisarealtwodimensional closed manifold.
A projective structure on a compact riemann surface c of genus g is given by an atlas with transition functions in pgl2,c. Mathematics and its applications soviet series, vol 16. R from the fiber product of e with itself to the trivial bundle with fiber r such that the restriction of k to each fibre over m is a nondegenerate bilinear map of vector spaces. The ahlforsbersteichmuller theory for arbitrary riemann surfaces is outlined. Matrix bundles and operator algebras over a finitely. We have attempted to give an informal presentation of the main results, addressed to a general audience. From the perspective of ab, it is the symplectic quotient cssg aflatg of the space css of semistable holomorphic structures on a. These are notes for my eight lectures given at the c. I would greately appreciate some explanation of this issue. The first part tackles the classification of vector bundles on algebraic curves.
The riemannroch theorem is one of the most essential theorems on riemann surfaces, or so i am told. More precisely, there is a onetoone correspondence between superholomorphic structures on vector bundles over super riemann surfaces and unitary connections satisfying certain curvature constraints. Democrats across the nation have threatened that they will abandon their final remaining shred of sanity should the republicancontrolled senate vote to confirm brett kavanaugh to the supreme court in the next few weeks. He then extended the whole theory to the algebraic context in any characteristic. Con tributions b y other p eople are set in sans serif and mark ed their initials. A remark on universal coverings of holomorphic families of riemann surfaces imayoshi, yoichi and. The description for this book, lectures on vector bundles over riemann surfaces. Matrix bundles and operator algebras over a finitely bordered. Geometric and analytic aspects of higgs bundle moduli spaces lecture 1 jan swoboda abstract. This theorem, proved by mumford 40 in the early 1960s, is the foundation not only of these lectures but of a vast amount of work on bundles over riemann surfaces. The work presented here was done in collaboration with m. The case of quaternionic vector bundles can be handled in the same way. The author also discusses the construction and elementary properties of the moduli spaces of stable bundles.
We prove an existence result for stable vector bundles with arbitrary rank on an algebraic surface, and determine the birational structure of certain moduli space of stable bundles on a rational ruled surface. Full text of schottky uniformization and vector bundles over riemann surfaces see other formats schottky uniformization and vector bundles over riemann surfaces carlos florentino abstract. Equivalently, a projective structure is given by a p1. Lectures on hyperbolic geometry riccardo benedetti. The main character of these lectures is a finitedimensional vector space, the. Properties of conformal and quasiconformal mappings are used to construct the bers holomorphic model for teichmuller space. The most important example is the tangent bundle, a real vector bundle.
Connections on vector bundles over super riemann surfaces. Vector bundles on a compact riemann surface x have two discrete, topolog. The author then generalizes the projective line results from chapter 2 to the case of a coherent analytic sheaf over an arbitary compact riemann surface. The riemann roch theorem is one of the most essential theorems on riemann surfaces, or so i am told. We give a necessary and sufficient condition for this map to be a submersion, when restricted to. Maximally unstable flat bundles are shown to arise in this way. Holomorphic vector bundles over compact complex surfaces. Whether youve loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. A ciliated fat graph is a graph with a fixed linear order of ends of edges at each vertex. Riemannroch for vector bundles mathematics stack exchange. Everyday low prices and free delivery on eligible orders. An introduction to the topology of the moduli space of.
Moduli of vector bundles on compact riemann surfaces by m. Moduli of riemann surfaces, transcendental aspects sources. In particular le potier constructs hilbertshgrothendieck schemes of vector bundles, and treats mumfords geometric invariant theory. We study a natural map from representations of a free group of rank g in gln, c, to holomorphic vector bundles of degree over a compact riemann surface x of genus g, associated with a schottky. Lectures on riemann surfaces download book pdf full. Finding ebooks booklid booklid download ebooks for free. It is shown that this moduli space can be obtained as a quotient of the space of graph connections by the poisson action of a lattice gauge group endowed with a poissonlie structure. The functor which associates the locally free sheaf e oxe to a vector bundle e on x is an equivalence of categories between the category. Lectures on hyperbolic geometry riccardo benedetti, carlo petronio focussing on the geometry of hyperbolic manifolds, the aim here is to provide an exposition of some fundamental results, while being as selfcontained, complete, detailed and unified as possible. Moduli stack of vector bundles 5 identities, the pullback functors f for f.
Im using the letter m to denote the base space of the vector bundle as a concession to the fact that in most. Vector bundles, locally free sheaves and divisors 30 4. More precisely, there is a onetoone correspondence between superholomorphic structures on vector bundles over super riemann surfaces and unitary connections satisfying certain curvature. Lectures on vector bundles on riemann surfaces pdf free. Roughly speaking, k gives a kind of dot product not necessarily symmetric or. This book is based on courses given at columbia university on vector bun dles 1988 and on the theory of algebraic surfaces 1992, as well as lectures in the park city lias mathematics institute on 4manifolds and donald son invariants.
It provides an introduction to the subject, as well as presenting methods used in the study of complex manifolds in the special case of complex dimension one. From democratic senators to lay voters, liberalleaning citizens have. Vector bundles on riemann surfaces and conformal field theory. We prove an existence result for stable vector bundles with arbitrary rank on an algebraic surface, and determine the birational structure of certain moduli space. 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. Otto forster, lectures on riemann surfaces, hershel m. A vector bundle of rank r on x is a scheme with a surjective. Higgs bundles i had the opportunity to give at heidelberg university in november and december 2014. Riemann surfaces university of california, berkeley. Degenerations of the moduli spaces of vector bundles on curves sources. Poisson structure on moduli of flat connections on riemann.
310 1098 1515 1420 674 589 474 359 107 1557 732 265 1443 1445 469 991 1590 1015 519 897 324 814 224 1491 1647 908 454 1091 1476 1288 1179 120 564 1020 1020 376