Twisted Fourier-Mukai transforms and bundles on non-Kahler elliptic surfaces

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In 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.