actions on compact Kaehler manifolds

Authors:
James B. Carrell and Andrew John Sommese

Journal:
Trans. Amer. Math. Soc. **276** (1983), 165-179

MSC:
Primary 32M05; Secondary 32C10, 32G05

MathSciNet review:
684500

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Abstract: Whenever acts holomorphically on a compact Kaehler manifold , the maximal torus of has fixed points. Consequently, has associated Bialynicki-Birula plus and minus decompositions. In this paper we study the interplay between the Bialynicki-Birula decompositions and the -action. A representative result is that the Borel subgroup of upper (resp. lower) triangular matrices in preserves the plus (resp. minus) decomposition and that each cell in the plus (resp. minus) decomposition fibres -equivariantly over a component of . We give some applications; e.g. we classify all compact Kaehler manifolds admitting a -action with no three dimensional orbits. In particular we show that if is projective and has no three dimensional orbit, and if Pic, then . We also show that if admits a holomorphic vector field with unirational zero set, and if is reductive, then is unirational.

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DOI:
https://doi.org/10.1090/S0002-9947-1983-0684500-X

Article copyright:
© Copyright 1983
American Mathematical Society