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Vector Bundles over Elliptic Surfaces

  • Robert Friedman
Chapter
  • 1.3k Downloads
Part of the Universitext book series (UTX)

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.

Keywords

Modulus Space Exact Sequence Vector Bundle Line Bundle Smooth Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Robert Friedman
    • 1
  1. 1.Department of MathematicsColumbia UniveristyNew YorkUSA

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