Stable bundles on positive principal elliptic fibrations

Abstract Let π −→ X be a principal elliptic fibration over a Kähler base X . We assume that the Kähler form on X is lifted to an exact form on M (such fibrations are called positive). Examples of these are regular Vaisman manifolds (in particular, the regular Hopf manifolds) and Calabi-Eckmann manifolds. Assume that dimM > 2. Using the KobayashiHitchin correspondence, we prove that all stable bundles on M are flat on the fibers of the elliptic fibration. This is used to show that all stable vector bundles on M take form L⊗ π∗B0, where B0 is a stable bundle on X , and L a holomorphic line bundle. For X algebraic this implies that all holomorphic bundles on M are filtrable (that is, obtained by successive extensions of rank-1 sheaves). We also show that all positive-dimensional compact subvarieties of M are pullbacks of complex subvarieties on X .