Holomorphic functions on bundles over annuli

Holomorphic Connections on Filtered Bundles over Curves

Let X be a compact connected Riemann surface and EP a holomorphic principal P–bundle over X , where P is a parabolic subgroup of a complex reductive affine algebraic group G. If the Levi bundle associated to EP admits a holomorphic connection, and the reduction EP ⊂ EP × P G is rigid, we prove that EP admits a holomorphic connection. As an immediate consequence, we obtain a sufficient condition...

Holomorphic 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 ...

Holomorphic Functions and Vector Bundles on Coverings of Projective Varieties

Let X be a projective manifold, ρ : X̃ → X its universal covering and ρ∗ : V ect(X) → V ect(X̃) the pullback map for the isomorphism classes of vector bundles. This article establishes a connection between the properties of the pullback map ρ∗ and the properties of the function theory on X̃. We prove the following pivotal result: if a universal cover of a projective variety has no nonconstant holo...

0 On Holomorphic Jet Bundles

Holomorphic Principal Bundles Over Elliptic Curves II: The Parabolic Construction

Let E be a smooth elliptic curve and let G be a simple complex algebraic group of rank r. We shall always assume that π1(G) is cyclic and that c is a generator. We shall freely identify c with the corresponding element in the center of the universal cover G̃ of G. A C∞ G-bundle ξ0 over E has a characteristic class c1(ξ0) ∈ H(E;π1(G)) ∼= π1(G) which determines ξ0 up to C ∞ isomorphism. The goal o...