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$ {\rm SL}(2,\,{\bf C})$ 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
DOI: https://doi.org/10.1090/S0002-9947-1983-0684500-X
MathSciNet review: 684500
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Abstract: Whenever $ G = SL(2,C)$ acts holomorphically on a compact Kaehler manifold $ X$, the maximal torus $ T$ of $ G$ has fixed points. Consequently, $ X$ has associated Bialynicki-Birula plus and minus decompositions. In this paper we study the interplay between the Bialynicki-Birula decompositions and the $ G$-action. A representative result is that the Borel subgroup of upper (resp. lower) triangular matrices in $ G$ preserves the plus (resp. minus) decomposition and that each cell in the plus (resp. minus) decomposition fibres $ G$-equivariantly over a component of $ {X^T}$. We give some applications; e.g. we classify all compact Kaehler manifolds $ X$ admitting a $ G$-action with no three dimensional orbits. In particular we show that if $ X$ is projective and has no three dimensional orbit, and if Pic$ (X) \cong {\mathbf{Z}}$, then $ X = C{{\mathbf{P}}^n}$. We also show that if $ X$ admits a holomorphic vector field with unirational zero set, and if $ \operatorname{Aut}_0(X)$ is reductive, then $ X$ is unirational.


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DOI: https://doi.org/10.1090/S0002-9947-1983-0684500-X
Article copyright: © Copyright 1983 American Mathematical Society

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