2 2 N ov 1 99 8 HOLOMORPHIC PRINCIPAL BUNDLES OVER ELLIPTIC CURVES

This paper, the first in a projected series of three, is concerned with the classification of holomorphic principal G-bundles over an elliptic curve, where G is a reductive linear algebraic group. The motivation for this study comes from physics. The F-theory/heterotic string duality predicts that, given an elliptically fibered Calabi-Yau manifold M of dimension n over a base space B, together with a stable E 8 × E 8-bundle over M and a so-called complexified Kähler class, there should be an associated Calabi-Yau manifold W of dimension n + 1, fibered over the same base B, where the fibers are elliptic K3 surfaces. In some sense, this paper and its two sequels are an attempt to understand this prediction in purely mathematical terms. The physical approach to this construction has been discussed in [14], as well as in the many references in that paper. For several mathematical reasons it has seemed worthwhile to consider not just E 8 × E 8 but the case of a general reductive group G. In this paper, we shall consider the problem of classifying holomorphic G-bundles over a single smooth elliptic curve E, as well as proving a number of auxiliary results which we shall need later. The methods of this paper are not well-suited to dealing with singular curves of arithmetic genus one or with families. Moreover, they do not work well in defining universal bundles, even locally in the case of a single smooth elliptic curve. In fact, all of these problems are already evident in the case of vector bundles with trivial determinant, i.e. in case G = SL n (C). In the first sequel to this paper, we will describe another method for constructing bundles, at least when G is simple, by considering deformations of certain minimally unstable G-bundles corresponding to special maximal parabolic subgroups of G. It turns out that this construction overcomes the problems outlined above, and it can be used to give a new proof and generalization of a theorem of Looijenga and Bernshtein-Shvartsman [24, 4] concerning the global structure of the moduli space of G-bundles. In Part III of this series, we specialize to the case motivated by physics and relate the case where the group G is E 6 , E 7 , E 8 to del Pezzo surfaces and simple elliptic singularities. The study of flat G-bundles over E also leads to a somewhat …