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Positive toric fibrations
Author(s) -
Verbitsky Misha
Publication year - 2009
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdn072
Subject(s) - subvariety , mathematics , pure mathematics , complex torus , fibered knot , toric variety , fiber bundle , invariant (physics) , principal bundle , torus , bundle , complex manifold , holomorphic function , vector bundle , geometry , mathematical physics , variety (cybernetics) , statistics , materials science , composite material
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 ⊂ 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.