Vector Bundles over Elliptic Surfaces

Abstract

This chapter gives a variety of techniques for understanding stable rank 2 vector bundles V over elliptic surfaces X. Because of constraints of space and time, we do not give complete proofs of all of the results, and this chapter is mainly intended as a sampler of various methods for studying bundles over X. For simplicity, we shall assume that X is simply connected, so that the base curve is ℙ¹ and shall concentrate on the case where X has a section v and the fibers are generic, i.e., smooth or nodal. After describing allowable elementary modifications for singular fibers, we begin with the technically much simpler case where the vector bundle V has degree 1 on every fiber. In this case, there is a unique stable bundle on each fiber, and correspondingly there is a unique stable rank 2 vector bundle V₀ on X, up to twisting by the pullback of a line bundle on the base curve, which restricts to a stable bundle on each fiber. The bundle V₀ then generates all stable bundles which have degree 1 on every fiber, via elementary modifications. A second approach to the moduli space is via sub-line bundles and extensions. We give a brief description of Donaldson invariants and some methods for computing them, and apply this to calculate the 2-dimensional Donaldson invariants coming from stable bundles of degree 1 on every fiber. Next we turn to the case where the degree of V on every fiber is 0. Here the moduli space of rank 2 semistable bundles of degree 0 on an elliptic curve is a ℙ¹ , and so the geometry of the moduli space when the fiber degree is 0 is much more complicated.