Vector Bundles on a Neighborhood of an Exceptional Curve and Elementary Transformations

Abstract

Let W be the germ of a smooth complex surface around an exceptional curve and let E be a rank 2 vector bundle on W. We study the cohomological properties of a finite sequence $E_i,1\le i\le t$ of rank 2 vector bundles canonically associated to E. We calculate numerical invariants of E in terms of the $E_i.$ If S is a compact complex smooth surface and E is a rank 2 bundle on the blow- up of S at a point, we show that all values of $c_2(E)-c_2({\pi }_{\ast }{E}^{\vee \vee })$ that are numerically possible are actually attained.

To appear in Forum Mathematicum