Holomorphic Line Bundles on the Loop Space of the Riemann Sphere
نویسنده
چکیده
The loop space LP1 of the Riemann sphere consisting of all C or Sobolev W k,p maps S → P1 is an infinite dimensional complex manifold. The loop group LPGL(2, C) acts on LP1. We prove that the group of LPGL(2, C) invariant holomorphic line bundles on LP1 is isomorphic to an infinite dimensional Lie group. Further, we prove that the space of holomorphic sections of these bundles is finite dimensional, and compute the dimension for a generic bundle.
منابع مشابه
The Picard Group of a Loop Space
The loop space LP1 of the Riemann sphere consisting of all C k or Sobolev W k,p maps S → P1 is an infinite dimensional complex manifold. We compute the Picard group Pic(LP1) of holomorphic line bundles on LP1 as an infinite dimensional complex Lie group with Lie algebra the Dolbeault group H(LP1). The group of Möbius transformations G and its loop group LG act on LP1. We prove that an element o...
متن کاملDolbeault Cohomology of a Loop Space
Loop spaces LM of compact complex manifolds M promise to have rich analytic cohomology theories, and it is expected that sheaf and Dolbeault cohomology groups of LM will shed new light on the complex geometry and analysis of M itself. This idea first occurs in [W], in the context of the infinite dimensional Dirac operator, and then in [HBJ] that touches upon Dolbeault groups of loop spaces; but...
متن کاملUniversal moduli spaces of surfaces with flat bundles and cobordism theory
For a compact, connected Lie group G, we study the moduli of pairs (Σ,E), where Σ is a genus g Riemann surface and E →Σ is a flat G-bundle. Varying both the Riemann surface Σ and the flat bundle leads to a moduli space Mg , parametrizing families Riemann surfaces with flat G-bundles. We show that there is a stable range in which the homology of Mg is independent of g. The stable range depends o...
متن کاملVector Bundles on Riemann Surfaces
1. Differentiable Manifolds 2 2. Complex Manifolds 3 2.1. Riemann Surfaces of Genus One 4 2.2. Constructing Riemann Surfaces as Curves in P 6 2.3. Constructing Riemann Surfaces as Covers 9 2.4. Constructing Riemann Surfaces by Glueing 10 3. Topological Vector Bundles 11 3.1. The Tangent and Cotangent Bundles 13 3.2. Interlude: Categories, Complexes and Exact Sequences 14 3.3. Metrics on Vector ...
متن کاملHolomorphic Fiber Bundles over Riemann Surfaces
For the purpose of this paper a fiber bundle F—>X over a Riemann surface X is meant to be a fiber bundle in the sense of N. Steenrod [62] where the base space is X, the fiber a complex space, the structure group G a complex Lie group that acts as a complex transformation group on the fiber, and the transition functions g%j{x) are holomorphic mappings into G. Correspondingly, cross-sections are ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2003