Positive toric fibrations

Abstract

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 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 $X\subset M$ 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 (math.AG/0403430). As follows from Borel-Remmert-Tits theorem, any compact simply connected 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.

21 pages, v3. The proof simplified, introduction added

References in corpus (2)

• Numerical characterization of the Kähler cone of a compact Kähler manifold
• Homogeneous HKT and QKT manifolds

Cited by in corpus (4)

• Complex geometry of moment-angle manifolds
• Subvarieties of hypercomplex manifolds with holonomy in SL(n,H)
• Canonical bundles of complex nilmanifolds, with applications to hypercomplex geometry
• Algebraic dimension and complex subvarieties of hypercomplex nilmanifolds