Stable bundles on positive principal elliptic fibrations (Q2388313)

Let \(M, X\) be compact and connected complex manifolds with \(X\) Kähler, \(\dim (M) \geq 3\), and \(\pi : M \to X\) a smooth holomorphic fibration such that there is an elliptic curve (considered as a complex Lie group) which acts holomorphically on \(M\) and freely and transitively on the fibers of \(\pi\) (a principal elliptic fibration). \(\pi\) is positive if there is a Kähler form \(\omega _X\) on \(X\) such that \(\pi ^\ast (\omega _X)\) is exact. Under this assumption the author gives results on the curvature of any Hermitian-Einstein bundle on \(M\) and prove (roughly) then any stable vector bundle on \(M\) is a twist by a line bundle of a bundle coming from \(X\). If \(X\) is projective, then he proves that every coherent sheaf on \(M\) is filtrable, i.e., it is obtained as successive extensions of rank one sheaves.