Unfiltrable Holomorphic Vector Bundles in a Cyclic Quotient of C \ {0} by E. Ballico

Let G := Z/nZ, n ≥ 2, act diagonally on C and set X := C 2 \ {0}/G (a punctured neighborhood for the normal surface singularity An−1). Here we prove the existence of a rank two holomorphic vector bundle on X without rank one subsheaves. 1. Unfiltrable Vector Bundles Let M be a reduced and irreducible complex space and E a rank two holomorphic vector bundle on M . We will say that E is unfiltrable if there is no rank one torsion free sheaf L such that there is a non-zero map L → E. In the case M smooth it is sufficient to use holomorphic line bundles L to test if E is unfiltrable. If M is Stein, then no rank two holomorphic vector bundle on M is unfiltrable by Theorem A of Cartan Serre. For example of pairs (M,E) with M smooth compact complex surface and E unfiltrable, see [2] and [3]. In [1] we proved the existence of unfiltrable rank two holomorphic vector bundles on C\{0}. If one is interested in the case in which M = Y/S with Y Stein manifold and S discrete in M , then unfiltrable bundles may exist only if dim(Y ) = 2 by an extension theorem due to Serre ([6]). The aim of this paper is to prove the following result. Received May 5, 2004 and in revised form September 7, 2004. AMS 2000 Subject Classification: 32L05, 32L010, 32S05.