Vector bundles on non-Kaehler elliptic principal bundles (Q381154)

This paper studies holomorphic vector-bundles on certain non-Kähler compact complex manifolds. The authors start with a compact complex manifold \(S\) with a principal elliptic bundle \(\pi:X\to S\) which is not topologically a product. When \(S\) is Kähler, this implies that \(X\) is non-Kähler. They then study the moduli space of relatively semistable rank \(n\) vector bundles on \(X\), using a twisted Fourier-Mukai transform and a spectral cover construction. The final result is that this moduli space is corepresented by the relative Douady space of length \(n\) and relative dimension \(0\) subspaces of the relative Jacobian.