Vector bundles on riemann surfaces and conformal field. Vector bundles on riemann surfaces and conformal field theory arnaud beauville introduction the main character of these lectures is a. Lectures on the moduli stack of vector bundles on a curve. Differential geometry of holomorphic vector bundles on a curve. In particular le potier constructs hilbertshgrothendieck schemes of vector bundles, and treats mumfords geometric invariant theory. Lectures on vector bundles on riemann surfaces pdf free. Moduli of riemann surfaces, transcendental aspects sources. Concerning the vector bundle theory, his famous papers, with m. Properties of conformal and quasiconformal mappings are used to construct the bers holomorphic model for teichmuller space. Lectures on vector bundles on riemann surfaces gunning r. 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. A vector bundle of rank r on x is a scheme with a surjective. Algebraic surfaces and holomorphic vector bundles robert. The riemannroch theorem is one of the most essential theorems on riemann surfaces, or so i am told.
Holomorphic bundles over elliptic manifolds sources. Line integral from vector calculus over a closed curve. A remark on universal coverings of holomorphic families of riemann surfaces imayoshi, yoichi and nishimura, minori, kodai. 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. The main character of these lectures is a finitedimensional vector space, the. An introduction to the topology of the moduli space of. More precisely, there is a onetoone correspondence between superholomorphic structures on vector bundles over super riemann surfaces and unitary connections satisfying certain curvature. 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. 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. 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. 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. 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. Algebraic moduli problems, moduli of vector bundles for.
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. In mathematics, vector bundles on algebraic curves may be studied as holomorphic vector bundles on compact riemann surfaces. It also deals quite a bit with noncompact riemann surfaces, but does include standard material on abels theorem, the abeljacobi map, etc. Everyday low prices and free delivery on eligible orders. 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. Poisson structure on moduli of flat connections on riemann. The work presented here was done in collaboration with m.
Full text of schottky uniformization and vector bundles. So, we obtain a sheaf of oxmodules e oxe over x, locally isomorphic to or x. The first part tackles the classification of vector bundles on algebraic curves. 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. A remark on universal coverings of holomorphic families of riemann surfaces imayoshi, yoichi and. Lectures on riemann surfaces download book pdf full.
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. Introduction to holomorphic vector bundles on compact riemann. 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. Download this book grew out of lectures on riemann surfaces given by otto forster at the universities of munich, regensburg, and munster. 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. The goal of these lectures was to acquaint researchers in 4manifold topology with the classification of algebraic surfaces and with methods for. 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. Finding ebooks booklid booklid download ebooks for free. Con tributions b y other p eople are set in sans serif and mark ed their initials. Geometric and analytic aspects of higgs bundle moduli spaces lecture 1 jan swoboda abstract. Lectures given in the minischool on moduli spaces at the banach center warsaw 2630 april 2005.
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. However, since his not commutative, one has to be careful when writing down formulas. The author also discusses the construction and elementary properties of the moduli spaces of stable bundles. Degenerations of the moduli spaces of vector bundles on curves sources. In chapter 4 the author proves the famous riemannroch theorem for the case of vector. 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. 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. 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. E m a vector bundle on m, then a metric on e is a bundle map k. Stable vector bundles on algebraic surfaces internet archive. 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. Holomorphic vector bundles over compact complex surfaces.
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. Lectures on vector bundles cambridge studies in advanced mathematics 54. The most important example is the tangent bundle, a real vector bundle. 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. 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.
Equivalently, a projective structure is given by a p1. Other readers will always be interested in your opinion of the books youve read. 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. This text deals with vector bundles over a curve x. Narasimhan1 and by himself2, inaugurated the modern theory of holomorphic vector bundles on compact riemann surfaces. This book grew out of lectures on riemann surfaces which the author gave at the universities of munich, regensburg and munster. 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. But there are many other interesting vector bundles. E of regular sections of e over u has a structure of module over the algebra oxu. More precisely, there is a onetoone correspondence between superholomorphic structures on vector bundles over super riemann surfaces and unitary connections satisfying certain curvature constraints. This book deserves very serious consideration as a text for anyone contemplating giving a course on riemann surfaces. The description for this book, lectures on vector bundles over riemann surfaces. Mathematics and its applications soviet series, vol 16. 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.
Quantum cohomology of the moduli space of stable bundles over a. 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. Vector bundles on riemann surfaces and conformal field theory. These are notes for my eight lectures given at the c. An introduction to the topology of the moduli space of stable. Construct a ranktwo vector bundle over the smooth quadric x. Riemann surfaces university of california, berkeley.
The ahlforsbersteichmuller theory for arbitrary riemann surfaces is outlined. Riemannroch for vector bundles mathematics stack exchange. On vector bundles over algebraic and arithmetic curves. 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. Complex vector bundles over compact riemann surfaces. Otto forster, lectures on riemann surfaces, hershel m. From democratic senators to lay voters, liberalleaning citizens have. I would greately appreciate some explanation of this issue. Roughly speaking, k gives a kind of dot product not necessarily symmetric or. Vector bundles on a compact riemann surface x have two discrete, topolog. 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.
He then extended the whole theory to the algebraic context in any characteristic. Lectures on hyperbolic geometry riccardo benedetti. 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. A ciliated fat graph is a graph with a fixed linear order of ends of edges at each vertex. Matrix bundles and operator algebras over a finitely bordered. Buy lectures on vector bundles over riemann surfaces. This book grew out of lectures on riemann surfaces given by otto forster at the universities of munich, regensburg and munster. Connections on vector bundles over super riemann surfaces. Differential geometry of complex vector bundles shoshichi. 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. From the perspective of ab, it is the symplectic quotient cssg aflatg of the space css of semistable holomorphic structures on a. Moduli of vector bundles on compact riemann surfaces by m. I would also recommend griffithss introduction to algebraic curves a beautiful text based on lectures.
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