Irreducible vector bundles on some elliptic non-Kahler threefolds

We study rank-2 vector bundles on non-Kähler threefolds π:X→Bπ:X→B, which are elliptic principal bundles with at least one non-zero Chern class over a complex surface BB with no curves. In this case, we prove that every rank-2 irreducible vector bundle on XX is a pull-back from BB up to a twist by a line bundle. These 2-vector bundles are, via the Kobayashi–Hitchin correspondence, solutions of the Yang–Mills equations on the threefold XX.