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. One is to replace the notion of a moduli space with that of a moduli stack. To obtain invariants of hyperkaehler x one finds that the holomorphic vector bundles must be hyper holomorphic. Singularities, representation of algebras, and vector bundles lambrecht, 1985.
Vector bundles can be approached from both an algebraic and a geometric perspective. Over riemann surfaces holomorphic vector bundles are a central part of the theory of the moduli space of flat connections. Pdf on the holomorphic rank2 vector bundles with trivial. Xon a variety xis a variety ewhich is in some sense a vector space internal to the category of. To each knot in the link one associates a holomorphic bundle over a holomorphic symplectic manifold x. The study of algebraic vector bundles of higher ranks began in 1957, when it was shown by a. The basic facts about holomorphic structures used here can be found in ko. Brinzanescu, v holomorphic vector bundles over compact complex surfaces. Im vaguely aware of the atiyah class, but are there other examples. Djvu pronounced deja vu is a digital document format with advanced. Use features like bookmarks, note taking and highlighting while reading algebraic surfaces and holomorphic vector bundles universitext. A course in computational algebraic number theory, henri cohen.
Manifolds, curves, and surfaces, marcel berger bernard. By serres gaga, the category of holomorphic vector bundles on a smooth complex projective variety x viewed as a complex manifold is equivalent to the category of algebraic vector bundles i. Use features like bookmarks, note taking and highlighting while reading algebraic surfaces and. On the moduli spaces of vector bundles on algebraic surfaces 179 such that f pn. Dolgachev,topics in classical algebraic geometry f r. To obtain invariants of hyperkaehler x one finds that the holomorphic vector bundles must be hyperholomorphic. The set of sheaves eon pn of pure dimension dwith re rand. A predecessor of maruyamas theorem is a lemma below called maruyamas lemma, which we translate in a serviceable fashion. The geometry of degree4 characteristic classes and of line bundles on loop. Vector bundles of rank 2 and linear systems on algebraic surfaces. A nullstellensatz with nilpotents and zariskis main lemma on holomorphic functions. Springer have made a bunch of books available for free, here are. Holomorphic vector bundles on almost complex manifolds.
Feb 21, 2000 to each knot in the link one associates a holomorphic bundle over a holomorphic symplectic manifold x. The study of line bundles on projective varieties is a classical problem in algebraic geometry cf. The goal of these lectures was to acquaint researchers in 4manifold topology with the classification of algebraic surfaces and with methods for describing moduli spaces of holomorphic bundles on algebraic surfaces with a view toward computing donaldson invariants. The fact that they allow to reconstruct the vector bundle is a standard and tedious technique of constructing algebraic varieties by glueing several pieces. Splitting principle for holomorphic vector bundles mathoverflow. L x be a holomorphic line bunlde over a complex manifold x. Holomorphic vector bundles on nonalgebraic tori of dimension. When one looks at nonalgebraic complex surfaces, one still has a notion of stability for holomorphic vector bundles with respect to gauduchon metrics on the surface and one gets the corresponding moduli spaces as open parts. An introduction to the topology of the moduli space of stable. On the moduli spaces of vector bundles on algebraic surfaces. Many of the results on vector bundles appear for the first time in book form, backed by many examples, both of surfaces and vector bundles, and over 100 exercises forming an integral part of the text. Topics in classical algebraic geometry algebraic surfaces. Featured on meta the q1 2020 community roadmap is on the blog. On vector bundles over algebraic and arithmetic curves.
E is called a holomorphic vector bundle over m if i are biholomorphic. L x be a holomorphic line bunlde x holomorphic section. Galois descent theory 265 exercises 274 10 classification of algebraic surfaces and of stable bundles 277 outline of the classification of surfaces 277 proof of castelnuovos theorem 284 the albanese map 288 proofs of the classification theorems for surfaces 290. 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. Im trying to understand what higgs bundles are, and the paper im reading starts off by taking a holomorphic vector bundle on a kahlereinstein manifold. Most of our studies are concerned with algebraic vector bundles e. The geometry of degree4 characteristic classes and of line bundles. Vector bundles in algebraic geometry enrique arrondo notes prepared for the first summer school on complex geometry villarrica, chile 79 december 2010 1. He then extended the whole theory to the algebraic context in any characteristic. Then we use it to study the pullbacks to x n and y n of bundles on p n. Vector bundles on riemann surfaces and conformal field theory arnaud beauville introduction the main character of these lectures is a.
Algebraic moduli problems, moduli of vector bundles for. Browse other questions tagged algebraicgeometry complexgeometry vectorbundles holomorphicbundles or ask your own question. Holomorphic vector bundles on nonalgebraic surfaces. Algebraic surfaces and holomorphic vector bundles pdf free. This book is a collection of survey articles by the main speakers at the symposium and presents to the mathematical world an overview of the key areas of research involving vector. 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.
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. Grothendieck that algebraic vector bundles on the projective line are direct sums of line bundles. The 1993 durham symposium on vector bundles in algebraic geometry brought together some of the leading researchers in the field to explore further these interactions. A holomorphic connection on a holomorphic vector bundle e over m is a rst order di erential operator d. Algebraic surfaces and holomorphic vector bundles robert.
Some applications of vector bundles in algebraic geometry. Since we now have the correspondence between vector bundles and points of the grassmannian, it is natural to ask if the kp system produces all the vector bundles on arbitrary algebraic curves. Concerning the vector bundle theory, his famous papers, with m. A gauge transformation is also a new choice of basis, and so can be handled similarly. For a holomorphic vector bundle v, the corresponding coherent analytic sheaf given by its local holomorphic sections will also be denoted byv.
Kop algebraic surfaces and holomorphic vector bundles av robert friedman pa. On projectivized vector bundles and positive holomorphic sectional curvature angelynn alvarez, gordon heier, and fangyang zheng abstract. The existence problem for holomorphic structures on vector bundles over nonalgebraic surfaces is in general still open. Vector bundles on riemann surfaces and conformal field theory. Some applications of vector bundles in algebraic geometry daniele faenzi email address.
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. Symmetric powers of vector bundles on curves 257 restriction theorems 260 appendix. As a side question, im curious what additional invariants one can construct for holomorphic vector bundles, which dont make sense for an ordinary complex vector bundle. Topics in classical algebraic geometry algebraic surfaces and. Vector bundle, algebraic encyclopedia of mathematics. They also have a nice discussion of slopestability and the stratification. From a geometric perspective, recall that an ndimensional vector bundle e. Every algebraic set in nspace is the intersection of n hypersurfaces. Algebraic surfaces and golomorphic vector bundles friedman. We generalize a construction of hitchin to prove that, given any compact kahler manifold m with positive holomorphic sectional curvature and any holomorphic vector bundle e over m, the projec. The top component of the stratification those bundles where the stabilizer group is as small as possible is the stack of semistable vector bundles. Vector bundles on riemann surfaces and conformal field.
The goal of these lectures was to acquaint researchers in. Vector bundles, locally free sheaves and divisors 30 4. The structure group then acts as a matrix transformation between vector components, and between bases in the opposite direction. X,l, or h0x,l, the set of all holomorphic sections of l. The manifold ms1,d m1,d equals a connected component of the picard variety pic. Rationality of moduli spaces of vector bundles on rational surfaces costa, laura and miroroig, rosa m. We have attempted to give an informal presentation of the main results, addressed to a general audience. Most of our studies are concerned with algebraic vector bundles e with e e 1. Algebraic surfaces and holomorphic vector bundles universitext kindle edition by friedman, robert. As far as i understand, being kahler only needs an almost complex manifold and being einstein is a condition on the ricci curvature and doesnt seem to imply the manifold is complex. Categories for the working mathematician, saunders mac lane.
Browse other questions tagged algebraic geometry complexgeometry vector bundles holomorphic bundles or ask your own question. Rationality of moduli spaces of vector bundles on rational surfaces costa, laura and miro. Algebraic surfaces and holomorphic vector bundles springerlink. Jun 14, 2005 the homotopy types of gauge groups of nonorientable surfaces and applications to moduli spaces theriault, stephen, illinois journal of mathematics, 20. From holomorphic functions to complex manifolds, klaus fritzsche hans grauert. A holomorphic line bundle is a rank one holomorphic vector bundle. For projective surfaces, schwarzenberger proved that a topological complex vector bundle admits a holomorphic algebraic structure if and only if its first chern class belongs to the neronseveri. A note on holomorphic vector bundles over complex tori, nagoya math. X,l, or h0x,l, the set of all holomorphic sections of l over x. In mathematics, vector bundles on algebraic curves may be studied as holomorphic vector bundles on compact riemann surfaces. Download it once and read it on your kindle device, pc, phones or tablets. Vector bundles of rank 2 and linear systems on algebraic. Algebraic surfaces and holomorphic vector bundles universitext 9780387983615.
Friedman, algebraic surfaces and holomorphic vector bundles fu w. Algebraic vector bundles on punctured affine spaces and. He then extended the whole theory to the algebraic context. The aim of the course was to give an introduction to algebraic stacks with a particular emphasis on moduli stacks of vector bundles on algebraic curves. First we define an invariant e for algebraic vector bundles on p n. Given an algebraic vector bundle e on p n, e e is a nonnegative integer.