The basic facts about holomorphic structures used here can be found in ko. A connection d on a holomorphic vector bundle e is compatible with the holomorphic structure if d0,1 if a connection he on a holomorphic vector bundle e is both hermitian and compatible with the holomorphic structure is called chern connection. By the way, the question may become more interesting if we make certain assumptions, e. Holomorphic vector bundles, stability, and gauge theory. Holomorphic vector bundles on homogeneous spaces sciencedirect. 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. A connection r in e is said to be compatible with the holomorphic structure in e if p0,1rss for all sections s of e. The theory studies complex structures which are near a given complex structure on a compact differentiable manifold. A holomorphic line bundle is a rank one holomorphic vector bundle. Algebraic surfaces and holomorphic vector bundles algebraic surfaces and holomorphic vector bundles by robert friedman. As soon as g is nonabelian the holomorphic affine connection for which rightinvariant vector fields are parallel have nonvanishing torsion and. The holomorphic structure on the tangent bundle of a complex manifold is guaranteed by the remark that the. Stable vector bundles and einsteinhermitian connections are also generalized.
Holomorphic vector bundles northwestern university. 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. Classically, the existence of a holomorphic structure on a complex vector bundle is also equivalent to one being able to choose holomorphic frames for trivializing the bundle locally. Use features like bookmarks, note taking and highlighting while reading algebraic surfaces and holomorphic vector bundles universitext. Holomorphic bundles i existence 20th october 2011 8 11. Branched holomorphic cartan geometries and calabiyau. Pdf paraholomorphic structures and the connections of. Assuming that all of its chern classes are still hodge, does it admit a holomorphic structure. The proof is nearly identical to that of proposition 1. Algebraic surfaces and holomorphic vector bundles books. It is assumed that we can, over small pieces of m, take bases that vary nicely with the parameter. Pdf flat partial connections and holomorphic structures. The tangent bundle tm is indeed a vector bundle, of rank dimm.
Curvature of vector bundles associated to holomorphic brations. Local structures on ane bundles over holomorphic tangent bundle. Flat partial connections and holomorphic structures in c. Download in pdf, epub, and mobi format for read it on your kindle device, pc, phones or tablets. Similarly, ruled and elliptic surfaces are discussed in detail, before the geometry of vector bundles over such surfaces is analysed. If e is a holomorphic vector bundle, the cohomology of e is defined to be the sheaf cohomology of. All the compact complex parallelizable manifolds admit a holomorphic affine structure connection given by the trivialization of the holomorphic tangent bundle by rightinvariant vector fields. A base point of eis a point p2xwhere the space fsp. Its prototypical examples were those in 33, 32 and 25 as mentioned above. Recall that, for a real vector bundle e m, a holomorphic structure is specified by a triple j e, j m. Local structures on ane bundles over holomorphic tangent bundle violeta zalutchi 1 abstract. The connection w is compatible with the complex structure if w e nl,op.
In ve2, it was shown that any bundle and any coherent sheaf. The fundamental equations for general decompositions of vector bundles with connection are studied in the case where some of the sub bundle are paraholomorphic in order to prove existence and. For a holomorphic vector bundle v, the corresponding coherent analytic sheaf given by its local holomorphic sections will also be denoted byv. Pdf flat partial connections and holomorphic structures in. Descargar algebraic surfaces and holomorphic vector bundles.
Moreover, the associated cc vector bundle always has the structure of a holomorphic vector bundle. The structure of the rest of this paper is as follows. Introduction hermitian vector bundles and dirac operators. Pdf differential geometry of holomorphic vector bundles on a curve. The holomorphic vector bundle eover mis called anchored if there exists a holomorphic vector bundle morphism e.
Vector bundles article pdf available in proceedings of the american mathematical society 733. Browse other questions tagged complexgeometry vector bundles holomorphic bundles or ask your own question. For complex manifold m and holomorphic vector bundle e. A connection r in e is said to be compatible with the holomorphic structure in e if p0,1rs. For dimcm2, geometry of holomorphic vector bundles is drastically di. Y v, together with the natural projection to x, is a vector bundle on x, called the pullback bundle. 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. Complex vector bundles that are not holomorphic mathoverflow. The 2sphere s2 has the structure of a complex manifold since s2 can be gotten by glueing two. A key observation here is atiyahbott 83, section 7, that a u n unprincipal connection induces a holomorphic structure on the associated complex vector bundle by taking the 0, 1 0,1part of the connection 1form as the dolbeault operator. Holomorphic bundles on diagonal hopf manifolds arxiv.
It is shown that a generalized holomorphic vector bundle on a compact nonorientable kahler manifold admits an einsteinhermitian connection if and only if it is polystable. Nonorientable manifolds, complex structures, and holomorphic. There is a oneone holomorphic vector bundles on homogeneous spaces 165 correspondence between ginvariant holomorphic bundle structures on p and extensions of p. In the next section we describe the procedure which allows us to map out the vacuum structure induced by bundle holomorphy on complex structure moduli space. This is why okonek and schneider write in their book p. A holomorphic structure on a topological complex vector bundle is by definition a maximal holomorphic atlas on it local trivialisations with holomorphic transition. Download algebraic surfaces and holomorphic vector bundles. One also has an analogous theory of deformations of holomorphic vector bundles on a compact complex manifold. Our approachde ning the kernel bundle rather than a trace bundle. On the other hand, there is a very rich theory of special connections on a fairly general class of bundles both vector and principal. Roughly speaking, a vector bundle is a vector space depending on a parameter which varies on a manifold m. E is called the bundle space or the total space, m is called the base space. Moreover, this bundle can be provided with a natural holomorphic structure. 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 robert. Complex vector bundles and complex manifolds definition 1. Divisors and line bundles department of mathematics. Deformations of complex structures and holomorphic vector. Complex manifolds and holomorphic vector bundles by l. Show that every holomorphic vector bundle over xarises in this way. A holomorphic connection on a holomorphic vector bundle e over. In this paper, by analogy with 1, we study the ane bundles taking as the base manifold the holomorphic bundle t0m of a complex 2 3 manifold, which in particular contains the 2. Its associated locally free sheaf ow is the same as. A complex vector bundle is a fiber bundle with fiber cn and structure group gln,c. This book covers the theory of algebraic surfaces and holomorphic vector bundles in an integrated manner. E e, where eis a holomorphic vector bundle anchored over m. Vector bundles with trivial determinant 225 even fiber degree and multiple fibers 237 exercises 240 9 bogomolovs inequality and applications 245 statement of the theorem 245 the theorems of bombieri and reider 248 the proof of bogomolovs theorem 252 symmetric powers of vector bundles on curves 257 restriction theorems 260.
Deformation of vector bundles with holomorphic connections. In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex. We generalize this to an adapted generalized holomorphic structure d on a vector bundle e of rank k in this section. A typical stable vector bundle in this situation is non. The answer is probably no, but i cannot think of any example. There are, in fact, several ways of doing this, but if p is irreducible, this procedure is. E is called a holomorphic vector bundle over m if i are biholomorphic. A complex connection on eh is a smooth splitting of 2. In particular, we have,,the space of global holomorphic sections of e.
The notion of a holomorphic vector bundle is extended to the context under consideration. Finsler structures on holomorphic lie algebroids arxiv. If a is a complex manifold and f the tangents of type 0, 1, the flat partial connections in a c vector bundle e are put in correspondence with the holomorphic structures in e. Differential geometry of holomorphic vector bundles on a curve. Vacuum varieties, holomorphic bundles and complex structure. Mbe a hermitian vector bundle with hermitian metric h. In particular, homogeneous line modules over the podle. Download it once and read it on your kindle device, pc, phones or tablets. Xbe a holomorphic vector bundle of rank kand denote by e the vector space of holomorphic sections of e. Paraholomorphic structures and the connections of vector bundles over paracomplex manifolds. Our starting point is to focus on classes of bundles where the complex structure dependence can be. We can think of unitary connections on a unbundle as giving the structure of a holomorphic vector bundle to the associated complex vector bundle. A complex vector bundle is a fiber bundle with fiber cn and structure group gln, c. Holomorphic bundles ii stability 1st november 2011 8 15.
Deformations of complex structures and holomorphic vector bundles. Algebraic surfaces and holomorphic vector bundles universitext kindle edition by friedman, robert. Adjusting the holomorphic structure of a vector bundle. Does this imply that eadmits a at connection that preserves the subbundle f. In discussing chern classes it is also necessary to consider the category of di. In order to discuss moduli of holomorphic vector bundles, it is essential to start with di.
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