Topic
Complex vector bundle
About: Complex vector bundle is a research topic. Over the lifetime, 172 publications have been published within this topic receiving 2290 citations.
Papers published on a yearly basis
Papers
TL;DR: In this paper, the authors present a self-contained proof of the periodicity theorem for the index problem for elliptic operators, assuming only basic facts from algebra and topology.
Abstract: where K(X) is the Grothendieck group (2) of complex vector bundles over X. The general theory of these K-groups, as developed in [1], has found many applications in topology and related fields. Since the periodicity theorem is the foundation stone of all this theory it seems desirable to have an elementary proof of it, and it is the purpose of this paper to present such a proof. Our proof will be strictly elementary. To emphasize this fact we have made the paper entirely self-contained, assuming only basic facts from algebra and topology. In particular we do not assume any knowledge of vector bundles or K-theory. We hope that, by doing this, we have made the paper intelligible to analysts who may be unacquainted with the theory of vector bundles but may be interested in the applications of K-theory to the index problem for elliptic operators [2]. We should point out in fact that our new proof of the periodicity theorem arose out of an attempt to understand the topological significance of elliptic boundary conditions. This aspect of the matter will be taken up in a subsequent paper.(3) In fact for the application to boundary problems we need not only the periodicity theorem but also some more precise results that occur in the course of our present proof.
83 citations
1 Jan 1985
TL;DR: In this article, the Kodaira-Nakano identity for holomorphic vector bundles on Kahler manifolds was derived, which is the same as the identity derived in this paper.
Abstract: In this chapter, we derive the Kodaira-Nakano identity (1.58) for holomorphic vector bundles on Kahler manifolds. We begin by describing our notation and reviewing some basic concepts from differential geometry.
78 citations
TL;DR: In this article, it was shown that the Wodzicki residue of an elliptic differential operator P acting on sections of a complex vector bundle E over a closed compact manifold M and the trace of the corresponding heat operator e − tP is the integral of the second coefficient of the heat kernel expansion of Δ up to a proportional factor.
76 citations
Posted Content•
TL;DR: In this article, a equivalence relation between connections on a complex vector bundle is defined between connections, and bundles equipped with such an equivalence class are called Structured Bundles, and their isomorphism classes form an abelian semi-ring.
Abstract: A equivalence relation, preserving the Chern-Weil form, is defined between connections on a complex vector bundle. Bundles equipped with such an equivalence class are called Structured Bundles, and their isomorphism classes form an abelian semi-ring. By applying the Grothedieck construction one obtains the ring K, elements of which, modulo a complex torus of dimension the sum of the odd Betti numbers of the base, are uniquely determined by the corresponding element of ordinary K and the Chern-Weil form. This construction provides a simple model of differential K-theory, c.f.Hopkins-Singer (2005), as well as a useful codification of vector bundles with connection.
72 citations
TL;DR: In this article, the authors considered 2-dimensional vector bundles over the 3-dimensional complex projective space P3 and showed that the structure group of a quaternion line-bundle over P3 admits a holomorphic structure.
Abstract: On a compact complex manifold X it is an interesting problem to compare the continuous and holomorphic vector bundles. The case of line-bundles is classical and is well understood in the framework of sheaf theory. On the other hand for bundles E with dimE>dimX we are in the stable topological range and one can use K-theory. Much is known in this direction, for example the topological and holomorphic K-groups of all complex projective spaces are isomorphic. This paper deals with what is perhaps the simplest case not covered by the methods indicated above. We shall consider 2-dimensional complex vector bundles over the 3-dimensional complex projective space P3. Our aim is to prove (1.1) Theorem. Every continuous 2-dimensional vector bundle over P3 admits a holomorphic structure. The corresponding result for P2 was proved by Schwarzenberger [13], but this falls into the class of stable problems. In particular 2-dimensional vector bundles over P2 are determined by their Chern classes c 1 , c 2 . This is no longer true on P3 and therein lies the main difficulty and also the interest of this paper. In fact Horrocks in [-10] has already constructed holomorphic (actually algebraic) bundles with arbitrary cl, c 2 subject only to the topologically necessary condition that c~c 2 be even [8; p. 166]. It is not hard to see that, topologically, there are at most two bundles on P3 with given cl, c 2 . The two possibilities arise because the homotopy group n 5 (U(2)) ~ n 5 (S 3) which classifies 2-dimensional bundles over S 6, and acts on the bundles over P3, has order 2. It turns out that there are two sharply different cases depending on the parity of c~. In w 2 we study the case of even c 1 . By tensoring with line-bundles one reduces to the case of cl =0 in which case the structure group is SU(2)~-Sp(1). We view our 2-dimensional complex vector bundle as a quaternion line-bundle and this simplifies the classification because quaternion line-bundles over P3 are already
68 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 3 |
| 2020 | 3 |
| 2019 | 5 |
| 2018 | 6 |
| 2017 | 4 |
| 2016 | 6 |