$\textrm {SL}(2, \textbf {C})$ actions on compact Kaehler manifolds
HTML articles powered by AMS MathViewer
- 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.References
- A. Białynicki-Birula, Some theorems on actions of algebraic groups, Ann. of Math. (2) 98 (1973), 480–497. MR 366940, DOI 10.2307/1970915
- A. Białynicki-Birula, On action of $\textrm {SL}(2)$ on complete algebraic varieties, Pacific J. Math. 86 (1980), no. 1, 53–58. MR 586868, DOI 10.2140/pjm.1980.86.53 A. Borel, Seminar on transformations, Ann. of Math. Studies, no. 46, Princeton Univ. Press, Princeton, N.J., 1961. J. B. Carrell, and R. M. Goresky, On the homology of projective varieties with $C^{\ast }$ action, preprint.
- James B. Carrell and Andrew John Sommese, $\textbf {C}^{\ast }$-actions, Math. Scand. 43 (1978/79), no. 1, 49–59. MR 523824, DOI 10.7146/math.scand.a-11762
- James B. Carrell and Andrew John Sommese, Some topological aspects of $\textbf {C}^{\ast }$ actions on compact Kaehler manifolds, Comment. Math. Helv. 54 (1979), no. 4, 567–582. MR 552677, DOI 10.1007/BF02566293 —, Generalization of a theorem of Horrocks, preprint.
- Akira Fujiki, On automorphism groups of compact Kähler manifolds, Invent. Math. 44 (1978), no. 3, 225–258. MR 481142, DOI 10.1007/BF01403162
- Takao Fujita, On the hyperplane section principle of Lefschetz, J. Math. Soc. Japan 32 (1980), no. 1, 153–169. MR 554521, DOI 10.2969/jmsj/03210153
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0 H. Hironaka, Bimeromorphic smoothing of a complex analytic space, Math. Inst. Warwick Univ., England, 1971.
- G. Horrocks, Fixed point schemes of additive group actions, Topology 8 (1969), 233–242. MR 244261, DOI 10.1016/0040-9383(69)90013-5
- David I. Lieberman, Compactness of the Chow scheme: applications to automorphisms and deformations of Kähler manifolds, Fonctions de plusieurs variables complexes, III (Sém. François Norguet, 1975–1977) Lecture Notes in Math., vol. 670, Springer, Berlin, 1978, pp. 140–186. MR 521918
- Toshiki Mabuchi, On the classification of essentially effective $\textrm {SL}(2;\textbf {C})$ $\times \textrm {SL}(2;\textbf {C})$-actions on algebraic threefolds, Osaka Math. J. 16 (1979), no. 3, 727–744. MR 551585
- Shigefumi Mori and Hideyasu Sumihiro, On Hartshorne’s conjecture, J. Math. Kyoto Univ. 18 (1978), no. 3, 523–533. MR 509496, DOI 10.1215/kjm/1250522508
- R. W. Richardson Jr., The variation of isotropy subalgebras for analytic transformation groups, Math. Ann. 204 (1973), 83–92. MR 377129, DOI 10.1007/BF01431491
- Andrew John Sommese, Extension theorems for reductive group actions on compact Kaehler manifolds, Math. Ann. 218 (1975), no. 2, 107–116. MR 393561, DOI 10.1007/BF01370814
- Andrew John Sommese, On manifolds that cannot be ample divisors, Math. Ann. 221 (1976), no. 1, 55–72. MR 404703, DOI 10.1007/BF01434964
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