Stable bundles on positive principal elliptic fibrations

Let $Ì\stackrel\pi \arrow X$ 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 {\bf 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 bundles on $M$ take form $L\otimes \pi^* B_0$, 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 $X$.

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Published 1 January 2005