Lecture notes geometry of manifolds mathematics mit. 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. Let e be a complex vector bundle over a smooth manifold x. Holomorphic connections and extension of complex vector. Complex and holomorphic vector bundles springerlink. Recall that a primary kodaira surface xis a principal elliptic bundle over some elliptic curve b. The demand for effectiveness imposes to work with some extrinsic choices of coordinates which, at the same time, should be forgotten in order that the mathematical content be intrinsic. Holomorphic vector bundles on holomorphically convex complex manifolds edoardo ballico abstract. The serre duality theorem for holomorphic vector bundles over a strongly pseudoconvex manifold itoh, mitsuhiro and saotome, takanari, tsukuba journal of mathematics, 2007. The parabolic construction friedman, robert and morgan, john w. Curvature of vector bundles associated to holomorphic brations by bo berndtsson abstract let lbe a semipositive line bundle over a k ahler manifold, x, bered over a complex manifold y. Let e be a holomorphic vector bundle of rank r over a complex manifold m. Mbe a hermitian vector bundle with hermitian metric h. We denote by g kv the grasmann manifold of kdimensional subspaces of v and by pv g 1v the projective space of v.
Descargar algebraic surfaces and holomorphic vector bundles. 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. We prove that every holomorphic vector bundle on a noncommutative twotorus t can be obtained by successive extensions from standard holomorphic bundles considered in 2. Nov 19, 2010 be a regular, surjective holomorphic map between complex manifolds such that for all t. Fundamental examples are the holomorphic tangent bundle of a complex manifold, and its dual, the holomorphic cotangent bundle. A holomorphic vector bundle has holomorphic transition functions, but a vector bundle cannot have.
Introduction to holomorphic vector bundles on compact. Classification of holomorphic vector bundles on noncommutative twotori a. Complex vector bundles that are not holomorphic mathoverflow. Curvature of vector bundles associated to holomorphic fibrations. A useful criterion for constructing holomorphic strutures on vector bundles is given by. Any homogeneous vector bundle e over an irreducible hermitian symmetric space arising from an irreducible representation of v is stable. In order to discuss moduli of holomorphic vector bundles, it is essential to start with di. E is called the bundle space or the total space, m is called the base space. Exact sequence vector bundle line bundle holomorphic function riemann sphere these keywords were added by machine and not by the authors.
When x is compact and without boundary then it is wellknown that if the zeroes of. The existence problem for holomorphic structures on vector bundles over nonalgebraic surfaces is, in general, still open. A holomorphic connection on eh is a holomorphic splitting of 2. L x be a holomorphic line bunlde over a complex manifold x. Roughly speaking, picx can be computed by the discrete part, measured by its image in h2x,z and the continuous part from the vector space h1x,o x. Holomorphic vector bundles on almost complex manifolds. Note what we are calling hermitian yang mills metrics, are called hermitian einstein metrics in this reference. Our holomorphic bundles over x come equipped with a connection. Pdf holomorphic vector bundles on primary kodaira surfaces.
Contents preface v introduction 1 1 curves on a surface 7 introduction 7 invariants of a surface 8 divisors on a surface 9 adjunction and arithmetic genus the riemannroch formula 15 algebraic proof of the hodge index theorem 16 ample and nef divisors 18. In mathematics, vector bundles on algebraic curves may be studied as holomorphic vector bundles on compact riemann 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. This holomorphic line bundle is called the tautologicalbundle. Vector bundle line bundle complex manifold holomorphic section curvature form these keywords were added by machine and not by the authors. Although our primary interest lies in holomorphic vector bundles, we begin this chapter with the study of connections in di. A complex connection on eh is a smooth splitting of 2.
X \ k a holomorphic vector bundle, equipped with a holomorphic relative connection along the fibres of. Similarly, ruled and elliptic surfaces are discussed in detail, before the geometry of vector bundles over such surfaces is analysed. In order to discuss moduli of holomorphic vector bundles, it is essential to start with differentiable complex vector bundles. Pdf document information annals of mathematics fine hall washington road princeton university princeton, nj 08544, usa phone. Assuming the bers are compact and nonsingular we prove that the hermitian vector bundle eover y whose bers over points yare the spaces of global sections over. Excellent references for these materials include the textbooks 5, 8. A holomorphic vector bundle is also a continuous vector bundle and a smooth vector bundle, for obvious reasons. X we denote by e the vector space of holomorphic sections.
We assume that excx 6 x and x is not a stein manifold. May 18, 2004 communicated by joachim cuntz abstract. Use features like bookmarks, note taking and highlighting while reading algebraic surfaces and holomorphic vector bundles universitext. It is aimed at graduate students who have had a thorough first year course in algebraic geometry at the level of hartshornes algebraic geometry, as well as more advanced graduate students and researchers in the areas of algebraic geometry, gauge thoery, or 4manifold topolgogy. L x be a holomorphic line bunlde x holomorphic section.
It is important to note that, given a topological or smooth vector bundle e, holomorphicity of eis additional data namely, a holomorphic atlas on e, not just a condition. Differential geometry of holomorphic vector bundles on a curve. Holomorphic vector bundles on primary kodaira surfaces. C to c, by using the fact that a holomorphic map s. In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold x such that the total space e is a complex manifold and the projection map. E is called a holomorphic vector bundle over m if i are biholomorphic.
In this section we explain how hermitian holomorphic line bundles. Let e be a holomorphic vector bundle over a connected complex projective manifold m. Vector bundle extensions, sheaf cohomology, and the heterotic standard model braun, volker, he, yanghui, ovrut, burt a. Dual of a holomorphic vector bundle mathematics stack. Homogeneous holomorphic vector bundles uc berkeley math. A holomorphic line bundle is a rank one holomorphic vector bundle. Pdf differential geometry of holomorphic vector bundles. Ii holomorphic and algebraic vector bundles thesis. The goal of these lectures was to acquaint researchers in. This will be the basis for many of the subsequent arguments. I found out this publication from my dad and i suggested this pdf to learn.
Holomorphic vector bundles on homogeneous spaces by u. Picx on x is the image of l under the boundary map c1. Divisors and line bundles jwr wednesday october 23, 2001 8. This is then clearly an even locally trivial holomorphic bration and there is a naturally. Algebraic surfaces and holomorphic vector bundles universitext kindle edition by friedman, robert. Hermitian vector bundles and the equidistribution of the. Hermitian einstein metrics on vector bundles and stability.
Global sections of holomorphic vector bundles stack exchange. Here we prove the existence of a holomorphic vector. On the narasimhanseshadri correspondence for real and. Holomorphic bundles ii stability 1st november 2011 8 15. Vector bundles, isoparametric functions and radon transforms on symmetric spaces nagatomo, yasuyuki and takahashi, masaro, osaka journal of mathematics, 2019. V which is not a vector bundle is an o mmodule and hence admits a torsion plus free decomposition. Algebraic surfaces and holomorphic vector bundles universitext. If eis a complex vector bundle over mand, is a hermitian metric on e, then. Vector bundles of rank one are called line bundles. Holomorphic line bundles in the absence of nonconstant holomorphic functions x.
We shall be concerned with vector bundles over x, i. A compact complex manifold m is called a hodge manifold if there exists a positive line bundle l over m. C on a compact complex manifold, we turn to the next best thing, holomorphic sections of line bundles i. Holomorphic vector bundles on the riemann sphere springerlink. Holomorphic principal bundles over elliptic curves ii. Insection3,weshall be interested in the much more involved classi. S2 of rank 2 for each k2z or a complex vector bundle of rank 1, since g 01. Curvature of vector bundles 535 another example of the situation in theorem 1. In this thesis, we introduce a notion of asymptotic stability for a holomorphic vector bundle with a global holomorphic section on a projective manifold. Pdf holomorphic rank two vector bundles on blowups.
Vector bundles over an elliptic curve 415 embedded biregularly in some projective space. Download and read free online algebraic surfaces and holomorphic vector bundles universitext. E is a holomorphic vector bundle of rank r with trivial pullback to w. An hermitian structure h in e can be expressed, in terms of a local holomorphic frame field s l9, s r of e, by a. For k 0, we denote by k the sheaf of kforms and by ke the sheaf of evalued k. The lie bracket operation of vector elds gives a lie algebra structure on the bers of all the. Bergman kernel and stability of holomorphic vector bundles. From now on by a vector bundle we shall mean a holomorphic vector bundle, and a curve will always be a smooth, complex projective curve. In discussing chern classes it is also necessary to consider the category of di. Find materials for this course in the pages linked along the left.
This book covers the theory of algebraic surfaces and holomorphic vector bundles in an integrated manner. This process is experimental and the keywords may be updated as the learning algorithm improves. Curvature of vector bundles associated to holomorphic. Relation to complex vector bundles with flat holomorphic connection. One could even talk about algebraic vector bundles, where v is an algebraic variety and the local trivialisations are morphisms of varieties from vto an. Download it once and read it on your kindle device, pc, phones or tablets. It is aimed at graduate students who have had a thorough firstyear course in algebraic. Introduction in 5,6,7 i introduced the concept of einsteinhermitian vector bundle. We prove in this paper that the equivalence classes of holomorphic vector bundles arising from ndimensional irreducible unitary representations of the. Complex vector bundles and complex manifolds definition 1.
Let us now pass on to the case of holomorphic vector bundles. Sure, it really is enjoy, continue to an amazing and interesting literature. Holomorphic vector bundles northwestern university. Moduli of real and quaternionic holomorphic vector bundles. Read and download ebook algebraic surfaces and holomorphic vector bundles universitext pdf public ebook library. Pdf uniqueness of balanced metrics on holomorphic vector. Cp1 c2 must be constant to show that l has no sections. Let e be a holomorphic sub bundle of a holomorphic vector bundle e over a manifold with hermitian metric. In this section by a hyper holomorphic vector bundle i will mean a. Algebraic surfaces and holomorphic vector bundles book. Curvature of vector bundles associated to holomorphic brations. In this section we explain how hermitian holomorphic line bundles carry a natural connection and hence one can talk of the curvature of such a bundle. The known paths to the production of holomorphic functions on x.
This is why okonek and schneider write in their book p. We prove that the special metric on the bundle studied by bradlow is the limit of a sequence of balanced metrics that are induced from the asymptotic stability. We write h for an hermtian metric on e,which of course induces. Let y be its lie algebra identified with a subalgebra of g so that f c y c g. Algebraic surfaces and holomorphic vector bundles robert. We solve this problem in the case of rank 2 vector bundles over k3 surfaces and in the case of vector bundles of arbitrary rank over all known surfaces of class vii. B is a complex manifold e together with a complex base b so that the transition data. Holomorphic vector bundles on a compact riemann surface. Two vector bundles over the grasmann g kv are the tautological bundle t. We begin by recalling the definition of a vector bundle. 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. The tangent bundle tm is indeed a vector bundle, of rank dimm. The holomorphic tangent bundle of a complex manifold. Holomorphic vector bundles on nonalgebraic surfaces.
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