2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/68438In this paper, we study holomorphic rank-2 vector bundles on non-K\" ahler elliptic surfaces. Our main tool for analysing these bundles is of course the spectral cover. However, given the non-Kähler condition, the elliptic surfaces we are considering do not have sections and gerbes naturally arise in this context. The spectral construction presented in this paper is a modification of the Fourier-Mukai transform for elliptic fibrations without a section. After examining some of the properties of this Fourier-Mukai transform, we give a complete classification of vector bundles on these surfaces.13 pagesAlgebraic GeometryComplex Variables14J60; 14D22, 14F05, 14J27, 32J15text