Rigid holomorphic rank 2 vector bundles on non-K\"ahler surfaces

The interest in rigid vector bundles (with respect to determinant preserving deformations) stems from various sources. From a geometric point of view, non-K\"ahler manifolds are of particular interest with respect to this problem. In this article, a description of the various possible cases of rigid rank 2 bundles on non-K\"ahler surfaces is presented. The largest gap of knowledge is for minimal class VII surfaces with $b_{2}(X)\ge 4$.