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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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$\textrm {SL}(2, \textbf {C})$ actions on compact Kaehler manifolds
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by James B. Carrell and Andrew John Sommese PDF
Trans. Amer. Math. Soc. 276 (1983), 165-179 Request permission

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 $\text {Pic}(X) \cong {\mathbf {Z}}$, 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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Additional Information
  • © Copyright 1983 American Mathematical Society
  • 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