SL(2, C)$ Actions on Compact Kaehler Manifolds

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 $...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 1983-03, Vol.276 (1), p.165-179, Article 165
Main Authors: Carrell, James B., Sommese, Andrew John
Format: Article
Language:English
Subjects:
Algebra
Algebraic conjugates
Isotropy
Lie groups
Mathematical manifolds
Mathematical theorems
Subalgebras
Tangents
Vector fields
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Summary: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 $\operatorname{Pic}(X) \cong Z$, then $X = CP^n$. 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.
ISSN:0002-9947
1088-6850
DOI:10.2307/1999424