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Equivariant completions of homogeneous algebraic varieties by homogeneous divisors. (English) Zbl 0537.14033
Let X be a smooth complete algebraic \({\mathbb{C}}\)-variety and G a connected linear group acting on X and having a Zariski open orbit \(\Omega\subset X\). By work of Borel \(A=X\backslash \Omega\) has at most 2 connected components. When A is connected classification of such actions was performed by the author in Math. USSR, Izv. 11, 293-307 (1977); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 41, 308-324 (1977; Zbl 0373.14016). In the present paper classification is performed when A is a connected divisor in X which is a homogeneous space for G; in this case \(X=\Omega \cup A\) is a two-orbit space. Main tool when G is semisimple is what the author calls the Mostow-Karpelevič fibration [see D. G. Mostow, Am. J. Math. 77, 247-278 (1955; Zbl 0067.160); and 84, 466-474 (1962; Zbl 0123.163); F. I. Karpelevich, Usp. Mat. Nauk 11, No.3, 131-138 (1956; Zbl 0072.182)]. The following geometric result is proved: any X as above can be fibered over a G-homogeneous complete variety with fibres isomorphic to a projective space, a product of projective spaces, a Grassmannian or a certain homogeneous space of \(E_ 6\); in particular such an X is projective and rational. Finally a classification of complete normal two orbit surfaces is provided.
Reviewer: A.Buium

MSC:
14M17 Homogeneous spaces and generalizations
14L30 Group actions on varieties or schemes (quotients)
14J10 Families, moduli, classification: algebraic theory
14C20 Divisors, linear systems, invertible sheaves
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[1] AHIEZER, D.N.: Algebraic groups acting transitively in the complement of a homogeneous hypersurface. Dokl.Akad.Nauk SSSR 245:2 281 - 284 (1979) (Russ.).Engl. Trans.: Soviet Math. Dokl. 20, 278 – 281 (1979)
[2] AHIEZER, D.N.: Dense orbits with two ends. Izv. Akad. Nauk SSSR, ser. mat., 41:2, 308 - 324 (1977) (Russ.). Engl. Trans.:Math. USSR, Izvestija 11, 293 – 307 (1977)
[3] BIALYNICKI-BIRULA, A.: On homogeneous affine spaces of linear algebraic groups. Amer. J. Math. 85:4, 577 - 582 (1963) · Zbl 0116.38202 · doi:10.2307/2373109
[4] BOREL, A.: Les bouts des espaces homogènes de groupes de Lie. Ann. of Math. 58:3, 443 - 457 (1953) · Zbl 0053.13002 · doi:10.2307/1969747
[5] BOREL, A.: Linear algebraic groups., New York - Amsterdam: W. A. Benjamin 1969 · Zbl 0206.49801
[6] BOREL, A., TITS, J.; Eléments unipotents et sous-groupes paraboliques de groupes réductifs. Inv. Math. 12:2, 95 - 104 (1971) · Zbl 0238.20055 · doi:10.1007/BF01404653
[7] BOTT, R.: Homogeneous vector bundles. Ann. of Math. 66:2, 203 - 248 (1957) · Zbl 0094.35701 · doi:10.2307/1969996
[8] BOURBAKI, N.: Groupes et algèbres de Lie, 2-ième partie. Paris: Hermann 1968
[9] BREDON, G.E.: Introduction to compact transformation groups. New York London: Academic Press 1972 · Zbl 0246.57017
[10] HUCKLEBERRY, A.T. , OEL JEKLAUS , E .: Homogeneous spaces from a complex analytic viewpoint (to appear)
[11] HUCKLEBERRY, A.T. , SNOW, D.: Almost homogeneous Kähler manifolds with hyper surface orbits (to appear) · Zbl 0507.32023
[12] KARPELEVIČ, F.I.: On a fibering of homogeneous spaces. Uspehl Mat. Nauk 11:3, 131 - 138 (1956) (Russ.),
[13] MONTGOMERY, D., YANG, C.T.: The existence of a slice. Ann. of Math. 65:1, 108 - 116 (1957) · Zbl 0078.16202 · doi:10.2307/1969667
[14] MOSTOW, G.D.: On covariant fiberings of Klein spaces I, II. Amer.J.Math. 77:2, 247 - 278 (1955); 84:3, 466 – 474 (1962) · Zbl 0067.16003 · doi:10.2307/2372530
[15] OELJEKLAUS, E.: Ein Hebbarkeitssatz für Automorphismengruppen kompakter komplexer Mannigfaltigkeiten. Math.Ann. 190:2, 154–166 (1970) · Zbl 0195.36902 · doi:10.1007/BF01431498
[16] POTTERS, J.: On almost homogeneous compact complex analytic surfaces. Inv.Math. 8:3, 244 - 266 (1969) · Zbl 0205.25102 · doi:10.1007/BF01406077
[17] SHAFAREVICH, I.R.: Basic algebraic geometry. Moscow: Nauka 1972 (Russ.) Engl.Translation- New York - Heidelberg - Berlin: Springer-Verlag (Grundlehren 213) 1974
[18] WANG, H. -C.: Two-point homogeneous spaces. Ann. of Math. 55:1, 177 - 191 (1952) · Zbl 0048.40503 · doi:10.2307/1969427
[19] WEISFEILER, B. Yu.: On a certain class of unipotent subgroups of semisimple algebraic groups. Uspehi Mat. Nauk. 21:2, 222–223 (1966) (Russ.)
[20] WOLF, J.A.: Space of constant curvature. New York: Mc Graw-Hill 1967 · Zbl 0162.53304
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