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 dim M > 2. Using the Kobayashi-Hitchin 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

more »

... es on M take form L ⊗ π * B 0 , where B 0 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. Proof. This is Theorem 4.3. For a definition of stability and Hermitian-Einstein connections see Section 3. Theorem 1.1 implies the following corollary. Proposition 1.2: Let T be an elliptic curve, and M π −→ X a positive principal T -fibration, equipped with a preferred Hermitian metric. The universal covering T acts on M in a standard way (its action is factorized through T ). Consider a stable bundle B on M . Then B is equipped with a natural holomorphicTequivariant structure. Proof. This is Proposition 4.6. A similar argument proves Proposition 1.3: Let M π −→ X be a positive principal elliptic fibration, and Z ⊂ M a closed positive-dimensional subvariety. Then Z = π * (Z 0 ) for some complex subvariety Z 0 ⊂ X. Proof. This is Proposition 4.5.