Vector bundles on non-Kaehler elliptic principal bundles

Abstract

Abstract. We study relatively semi-stable vector bundles and their moduli on non-Kähler principal elliptic bundles over compact complex manifolds of arbitrary dimension. The main technical tools used are the twisted FourierMukai transform and a spectral cover construction. For the important example of such principal bundles, the numerical invariants of a 3-dimensional nonKähler elliptic principal bundle over a primary Kodaira surface are computed.

Algebraic methods for vector bundles on non-Kähler elliptic fibrations

“…Note that by flatness and compatibility with restriction to fibres, C(F T ) depends only on the equivalence class of F T . We are now able to present the theorem which generalises Theorem 8, see [26].…”

“…The relative Jacobian exists also in our non-Kähler case. It is just the product S × E * and has the following special properties (see [26]):…”

“…The following theorem (see [26]) supplies us with the main tool for the treatment of the moduli spaces M X (n, 0) of relatively semistable vector bundles on X of rank n and degree 0 on the fibres X s (for vector bundles on elliptic curves see [3], [61]). It is an analog of the Theorem 6.5.4 [30] (see also [31]): Theorem 10.…”

“…The spectral cover C(F) of F is the 0−th Fitting subscheme (see [54], [39]) of J given by the Fitting ideal sheaf F itt 0 (Φ 1 (F)) of Φ 1 (F). For details see [26].…”

“…For the proof, see [26]. The spectral covers C(F T ) lead us to consider the relative Douady functors D n : (An/S) op → (Sets),…”