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Journal of the London Mathematical Society Advance Access originally published online on January 6, 2009
Journal of the London Mathematical Society 2009 79(2):294-308; doi:10.1112/jlms/jdn072
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© 2009 London Mathematical Society

Positive toric fibrations

Misha Verbitsky

Institute of Theoretical and Experimental Physics
B. Cheremushkinskaya 25
Moscow 117259
Russia
verbit@mccme.ru

A principal toric bundle M is a complex manifold equipped with a free holomorphic action of a compact complex torus T. Such a manifold is fibred over M/T, with fibre T. We discuss the notion of positivity in fibre bundles and define positive toric bundles. Given an irreducible complex subvariety X sub M of a positive principal toric bundle, we show that either X is T-invariant, or it lies in an orbit of a T-action. For principal elliptic bundles, this theorem is known from Verbitsky [Math. Res. Lett. 12 (2005) 251–264]. As follows from the 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.

2000 Mathematics Subject Classification 22E15, 32J25, 14C99.

Received March 7, 2007; revised August 27, 2008; published online January 6, 2009.


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