Abstract
Let X be an analytic compact connected complex variety. Denote by \(\mathcal{M}(X)\) the field of global meromorphic functions on X. About 1950 it was shown that the transcendence degree of \(\mathcal{M}(X)\)over C is less than or equal to the dimension of X ([Si], [GA] p.63). Those varieties for which equality holds we call Artin-Moišezon varieties; they were extensively studied around the end of the 1960’s ([M], [A], … ). They have an “algebraic structure” which generalizes the notion of algebraic structure on classical algebraic varieties (i.e. proper smooth schemes of finite type over C). Although at one time it was not clear that nonprojective ArtinMoišezon varieties existed ([CK], [Hi] Problem 34, [Nag]), there are by now many examples ([Ha] Appendix B, [M-S]), and the term “algebraic variety” is often used synonymously with Artin-Moišezon variety.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. Artin, Algebraization of formal moduli: II. Existence of modifications, Ann. of Math. 91 (1970), 88–135
A. Bialynicki-Birula, On the action of SL2 on complete algebraic varieties, Pacific J. Math. 86 (1980), 53–58
M. Brion, D. Luna, Th. Vust, Espaces homogènes sphériques, Invent. Math. 84 (1986), 617–632
H. Cartan, Quotient d’un espace analytique par un groupe d’automor-phismes. In: Algebraic geometry and topology: a symposium in honor of S. Lefschetz, Princeton Univ. Press, 1957
W.L. Chow, K. Kodaira, On analytic surfaces with two independent meromorphic functions, Proc. N.A.S. USA 38 (1952), 319–325
P. Griffiths, J. Adams, Topics in Algebraic and Analytic Geometry, Princeton Univ. Press, 1974
R. Hartshorne, Algebraic Geometry, Springer-Verlag, 1977
F. Hirzebruch, Some problems on differentiable and complex manifolds, Ann. of Math. 60 (1954), 213–236
D. Luna, Th. Vust, Plongements d’espaces homogènes, Comment. Math. Helv. 58 (1983), 186–245
B.G. Moisezon, On n-dimensional compact varieties with n algebraically independent meromorphic functions I, II, III, Amer. Math. Soc. Translations, ser. 2, 63 (1967)
L. Moser-Jauslin, Normal SL2 /Γ-embeddings, Thesis, Univ. of Geneva (1987)
L. Moser-Jauslin, Smooth Embeddings of SL2 and PGL2, to appear
S. Müller-Stach, Bimeromorphe Geometrie von dreidimensionalen Moišezon Mannigfaltigkeiten, Diplomarbeit, Universität Bayreuth (1987)
S. Mukai, H. Umemura, Minimal rational threefolds. In: Algebraic Geometry, Lecture Notes in Math. 1016, Springer-Verlag, 1983
M. Nagata, On the embedding problem of abstract varieties in projective varieties, Mem. Coll. Sci. Univ. Kyoto, Ser A, Math. 30 (1956), 71–82
T. Nakano, On equivariant completions of three-dimensional homogeneous spaces of SL2(C), to appear in Jap. J. of Math
C.L. Siegel, Meromorphe Funktionen auf kompakten analytischen Mannigfaltigkeiten, Nach. Akad. Wiss. 4, Göttingen (1955), 71–77
S. Sumihiro, Equivariant Completion, J. Math. Kyoto Univ. 14 (1974), 1–28
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1989 Birkhäuser Boston, Inc.
About this chapter
Cite this chapter
Luna, D., Moser-Jauslin, L., Vust, T. (1989). Almost Homogeneous Artin-MoiŠezon Varieties Under the Action of PSL2(C). In: Kraft, H., Petrie, T., Schwarz, G.W. (eds) Topological Methods in Algebraic Transformation Groups. Progress in Mathematics, vol 80. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-3702-0_7
Download citation
DOI: https://doi.org/10.1007/978-1-4612-3702-0_7
Publisher Name: Birkhäuser Boston
Print ISBN: 978-1-4612-8219-8
Online ISBN: 978-1-4612-3702-0
eBook Packages: Springer Book Archive