Positive toric fibrations

A principal toric bundleM is a complex manifold equipped with a free holomorphic action of a compact complex torus T . Such a manifold is fibered over M/T , with fiber T . We discuss the notion of positivity in fiber bundles and define positive toric bundles. Given an irreducible complex subvariety X ⊂ M of a positive principal toric bundle, we show that either X is T -invariant, or it lies in an orbit of T -action. For principal elliptic bundles, this theorem is known ([V]). As follows from Borel-Remmert-Tits theorem, any simply connected compact homogeneous complex manifold is a principal toric bundle. We show that compact Lie groups with left-invariant complex structure I are positive toric bundles, if I is generic. Other examples of positive toric bundles are discussed.