HOLOMORPHIC BUNDLES OVER ALGEBRAIC CURVES AND NON-LINEAR EQUATIONS

Vector bundles over algebraic curves

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 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...

Holomorphic Principal Bundles Over Elliptic Curves III: Singular Curves and Fibrations

Let G be a reductive complex linear algebraic group, which in this paper for simplicity we shall always assume to be simple and simply connected, and let Lie(G) = g. This paper is the third in a series concerned with the study of holomorphic principal G-bundles over elliptic fibrations. The first paper in the series studied the moduli space M of Gbundles on a smooth elliptic curve E by means of...

Holomorphic bundles on O(−k) are algebraic

We show that holomorphic bundles on O(−k) for k ≥ 0 are algebraic. We also show that such bundles are trivial outside the zero section. 1 Preliminaires The line bundle on P given by transition function z is usually denoted O(−k). Since we will be studying bundles over this space, we will denote O(−k) by Mk when we want to view this space as the base of a bundle. We give Mk the following charts ...