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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[**A. Białynicki-Birula,**B-B**]*Some theorems on actions of algebraic groups*, Ann. of Math. (2)**98**(1973), 480–497. MR**0366940****[**A. Białynicki-Birula,**B-B**]*On action of 𝑆𝐿(2) on complete algebraic varieties*, Pacific J. Math.**86**(1980), no. 1, 53–58. MR**586868****[**A. Borel,**Bo**]*Seminar on transformations*, Ann. of Math. Studies, no. 46, Princeton Univ. Press, Princeton, N.J., 1961.**[**J. B. Carrell, and R. M. Goresky,**C-G**]*On the homology of projective varieties with**action*, preprint.**[**James B. Carrell and Andrew John Sommese,**C-S**]*𝐶*-actions*, Math. Scand.**43**(1978/79), no. 1, 49–59. MR**523824****[**James B. Carrell and Andrew John Sommese,**C-S**]*Some topological aspects of 𝐶* actions on compact Kaehler manifolds*, Comment. Math. Helv.**54**(1979), no. 4, 567–582. MR**552677**, 10.1007/BF02566293**[**-,**C-S**]*Generalization of a theorem of Horrocks*, preprint.**[**Akira Fujiki,**F**]*On automorphism groups of compact Kähler manifolds*, Invent. Math.**44**(1978), no. 3, 225–258. MR**0481142****[**Takao Fujita,**Fu**]*On the hyperplane section principle of Lefschetz*, J. Math. Soc. Japan**32**(1980), no. 1, 153–169. MR**554521**, 10.2969/jmsj/03210153**[**Robin Hartshorne,**H**]*Algebraic geometry*, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR**0463157****[**H. Hironaka,**Hi**]*Bimeromorphic smoothing of a complex analytic space*, Math. Inst. Warwick Univ., England, 1971.**[**G. Horrocks,**Ho**]*Fixed point schemes of additive group actions*, Topology**8**(1969), 233–242. MR**0244261****[**David I. Lieberman,**L**]*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,**M**]*On the classification of essentially effective 𝑆𝐿(𝑛;𝐶)-actions on algebraic 𝑛-folds*, Osaka J. Math.**16**(1979), no. 3, 745–758. MR**551586****[**Shigefumi Mori and Hideyasu Sumihiro,**M-S**]*On Hartshorne’s conjecture*, J. Math. Kyoto Univ.**18**(1978), no. 3, 523–533. MR**509496****[**R. W. Richardson Jr.,**R**]*The variation of isotropy subalgebras for analytic transformation groups*, Math. Ann.**204**(1973), 83–92. MR**0377129****[**Andrew John Sommese,**S**]*Extension theorems for reductive group actions on compact Kaehler manifolds*, Math. Ann.**218**(1975), no. 2, 107–116. MR**0393561****[**Andrew John Sommese,**S**]*On manifolds that cannot be ample divisors*, Math. Ann.**221**(1976), no. 1, 55–72. MR**0404703**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
32M05,
32C10,
32G05

Retrieve articles in all journals with MSC: 32M05, 32C10, 32G05

Additional Information

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

Article copyright:
© Copyright 1983
American Mathematical Society