Positive toric fibrations

Details Positive toric fibrations Positive toric fibrations

2007-03-06

arxiv.org/abs/math/0703162v3

J. Lond. Math. Soc. (2) 79 (2009), no. 2, 294--308.

Mathematics

Algebraic Geometry

21 pages, v3. The proof simplified, introduction added

Scientific paper

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.

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