actions on compact Kaehler manifolds
Authors:
James B. Carrell and Andrew John Sommese
Journal:
Trans. Amer. Math. Soc. 276 (1983), 165179
MSC:
Primary 32M05; Secondary 32C10, 32G05
MathSciNet review:
684500
Fulltext 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 BialynickiBirula plus and minus decompositions. In this paper we study the interplay between the BialynickiBirula 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.
 [BB]
A.
BiałynickiBirula, Some theorems on actions of algebraic
groups, Ann. of Math. (2) 98 (1973), 480–497.
MR
0366940 (51 #3186)
 [BB]
A.
BiałynickiBirula, On action of 𝑆𝐿(2) on
complete algebraic varieties, Pacific J. Math. 86
(1980), no. 1, 53–58. MR 586868
(81i:14031)
 [Bo]
A. Borel, Seminar on transformations, Ann. of Math. Studies, no. 46, Princeton Univ. Press, Princeton, N.J., 1961.
 [CG]
J. B. Carrell, and R. M. Goresky, On the homology of projective varieties with action, preprint.
 [CS]
James
B. Carrell and Andrew
John Sommese, 𝐶*actions, Math. Scand.
43 (1978/79), no. 1, 49–59. MR 523824
(80h:32053)
 [CS]
James
B. Carrell and Andrew
John Sommese, Some topological aspects of 𝐶* actions on
compact Kaehler manifolds, Comment. Math. Helv. 54
(1979), no. 4, 567–582. MR 552677
(80m:32032), http://dx.doi.org/10.1007/BF02566293
 [CS]
, Generalization of a theorem of Horrocks, preprint.
 [F]
Akira
Fujiki, On automorphism groups of compact Kähler
manifolds, Invent. Math. 44 (1978), no. 3,
225–258. MR 0481142
(58 #1285)
 [Fu]
Takao
Fujita, On the hyperplane section principle of Lefschetz, J.
Math. Soc. Japan 32 (1980), no. 1, 153–169. MR 554521
(81c:14005), http://dx.doi.org/10.2969/jmsj/03210153
 [H]
Robin
Hartshorne, Algebraic geometry, SpringerVerlag, New
YorkHeidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
(57 #3116)
 [Hi]
H. Hironaka, Bimeromorphic smoothing of a complex analytic space, Math. Inst. Warwick Univ., England, 1971.
 [Ho]
G.
Horrocks, Fixed point schemes of additive group actions,
Topology 8 (1969), 233–242. MR 0244261
(39 #5578)
 [L]
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
(80h:32056)
 [M]
Toshiki
Mabuchi, On the classification of essentially effective
𝑆𝐿(𝑛;𝐶)actions on algebraic
𝑛folds, Osaka J. Math. 16 (1979),
no. 3, 745–758. MR 551586
(81k:14033b)
 [MS]
Shigefumi
Mori and Hideyasu
Sumihiro, On Hartshorne’s conjecture, J. Math. Kyoto
Univ. 18 (1978), no. 3, 523–533. MR 509496
(80j:14033)
 [R]
R.
W. Richardson Jr., The variation of isotropy subalgebras for
analytic transformation groups, Math. Ann. 204
(1973), 83–92. MR 0377129
(51 #13302)
 [S]
Andrew
John Sommese, Extension theorems for reductive group actions on
compact Kaehler manifolds, Math. Ann. 218 (1975),
no. 2, 107–116. MR 0393561
(52 #14370)
 [S]
Andrew
John Sommese, On manifolds that cannot be ample divisors,
Math. Ann. 221 (1976), no. 1, 55–72. MR 0404703
(53 #8503)
 [BB]
 A. BialynickiBirula, Some theorems on actions of algebraic groups, Ann. of Math. 98 (1973), 480497. MR 0366940 (51:3186)
 [BB]
 , On action of on complete algebraic varieties, Pacific J. Math. 86 (1980), 5358. MR 586868 (81i:14031)
 [Bo]
 A. Borel, Seminar on transformations, Ann. of Math. Studies, no. 46, Princeton Univ. Press, Princeton, N.J., 1961.
 [CG]
 J. B. Carrell, and R. M. Goresky, On the homology of projective varieties with action, preprint.
 [CS]
 J. B. Carrell and A. J. Sommese, actions, Math. Scand. 43 (1978), 4959. MR 523824 (80h:32053)
 [CS]
 , Some topological aspects of actions on compact Kaehler manifolds, Comment. Math. Helv. 54 (1979), 567582. MR 552677 (80m:32032)
 [CS]
 , Generalization of a theorem of Horrocks, preprint.
 [F]
 A. Fujiki, On automorphism groups of compact Kaehler manifolds, Invent. Math. 44, (1978), 225258. MR 0481142 (58:1285)
 [Fu]
 T. Fujita, On the hyperplane section principle of Lefschetz, J. Math. Soc. Japan 32 (1980), 153169. MR 554521 (81c:14005)
 [H]
 R. Hartshorne, Algebraic geometry, SpringerVerlag, New York, 1977. MR 0463157 (57:3116)
 [Hi]
 H. Hironaka, Bimeromorphic smoothing of a complex analytic space, Math. Inst. Warwick Univ., England, 1971.
 [Ho]
 G. Horrocks, Fixed point schemes of additive group actions, Topology 8 (1969), 233242. MR 0244261 (39:5578)
 [L]
 D. Lieberman, Compactness of the Chow scheme: applications to automorphisms and deformations of Kaehler manifolds, Séminairé Norquet, Lecture Notes in Math., vol. 670, SpringerVerlag, Berlin and New York, 1975. MR 521918 (80h:32056)
 [M]
 T. Mabuchi, On the classification of essentially effective actions on algebraic folds, Osaka J. Math. 16 (1979), 745758. MR 551586 (81k:14033b)
 [MS]
 S. Mori and H. Sumihiro, On Harthshorne's conjecture, J. Math. Kyoto Univ. 18 (1978), 523533. MR 509496 (80j:14033)
 [R]
 R. W. Richardson, Jr., The variation of isotropy subalgebras for analytic transformation groups, Math. Ann. 204 (1973), 8392. MR 0377129 (51:13302)
 [S]
 A. J. Sommese, Extension theorems for reductive group actions on compact Kaehler manifolds, Math. Ann. 218 (1975), 107116. MR 0393561 (52:14370)
 [S]
 , On manifolds that cannot be ample divisors, Math. Ann. 221 (1976), 5572. MR 0404703 (53:8503)
Similar Articles
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/S0002994719830684500X
PII:
S 00029947(1983)0684500X
Article copyright:
© Copyright 1983
American Mathematical Society
