Abstract
These are expanded notes from lectures on the geometry of spherical varieties given in Sanya. We review some aspects of the geometry of spherical varieties. We first describe the structure of B-orbits. Using the local structure theorems, we describe the Picard group and the group of Weyl divisors and give some necessary conditions for smoothness. We later on consider B-stable curves and describe in details the structure of the Chow group of curves as well as the pairing between curves and divisors. Building on these results we give an explicit B-stable canonical divisor on any spherical variety.
Similar content being viewed by others
References
Achinger, P., Perrin, N.: Spherical multiple flags. Advances Studies in Pure Math., 71, 53–74 (2016)
Bender, M., Perrin, N.: Singularities of closures of spherical B-conjugacy classes of nilpotent orbits. Preprint arXiv:1412.5654
Bia-lynicki-Birula, A.: Some theorems on actions of algebraic groups. Ann. of Math., 98 (2), 480–497 (1973)
Borel, A.: Linear algebraic groups. Second edition. Graduate Texts in Mathematics, 126. Springer-Verlag, New York, 1991
Brion, M.: Groupe de Picard et nombres caractéristiques des variétés sphériques. Duke Math. J., 58 (2), 397–424 (1989)
Brion, M.: Variétés sphériques et théorie de Mori. Duke Math. J., 72 (2), 369–404 (1993)
Brion, M.: Curves and divisors in spherical varieties. Algebraic groups and Lie groups, 21–34, Austral. Math. Soc. Lect. Ser., 9, Cambridge Univ. Press, Cambridge, 1997
Brion, M.: Equivariant Chow groups for torus actions. Transform. Groups, 2 (3), 225–267 (1997)
Brion, M.: Multiplicity-free subvarieties of flag varieties. Commutative algebra (Grenoble/Lyon, 2001), 13–23, Contemp. Math., 331, Amer. Math. Soc., Providence, RI, 2003
Brion, M.: Algebraic group actions on normal varieties. Preprint arXiv:1703.09506
Brion, M., Pauer, F.: Valuations des espaces homogènes sphériques. Comment. Math. Helv., 62 (2), 265–285 (1987)
Demazure, M.: Sous-groupes algébriques de rang maximum du groupe de Cremona. Ann. Sci. ENS, 3, 507–588 (1970)
Ewald, G., Wessels, U.: On the ampleness of invertible sheaves in complete projective toric varieties. Results Math., 19(3–4), 275–278 (1991)
Fulton, W.: Introduction to toric varieties. Annals of Mathematics Studies, 131. The William H. Roever Lectures in Geometry. Princeton University Press, Princeton, NJ, 1993
Fulton, W.: Intersection theory. Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 2. Springer-Verlag, Berlin, 1998
Fulton, W., MacPherson, R., Sottile, F., et al.: Intersection theory on spherical varieties. J. Algebraic Geom., 4 (1), 181–193 (1995)
Gandini, J.: Embeddings of spherical homogeneous spaces. Submitted to Acta Math. Sin., Engl. Ser.
Gandini, J., Pezzini, G.: Orbits of strongly solvable spherical subgroups on the flag variety. To appear in J. Algebraic Combin.
Gonzales, R., Pech, C., Perrin, N., et al.: Geometry of rational curves on horospherical varieties of Picard rank one. In preparation
Grosshans, F. D.: The invariants of unipotent radicals of parabolic subgroups. Invent. Math., 73, 1–9 (1983)
Grosshans, F. D.: Algebraic Homogeneous Spaces and Invariant Theory. Lecture Notes in Mathematics 1673, Springer-Verlag, Berlin, 1997
Hartshorne, R.: Algebraic geometry. Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977
Knop, F.: The Luna–Vust theory of spherical embeddings. Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989), 225–249, Manoj Prakashan, Madras, 1991
Knop, F.: On the set of orbits for a Borel subgroup. Comment. Math. Helv., 70 (2), 285–309 (1995)
Lazarsfeld, R.: Positivity in algebraic geometry. I. Classical setting: line bundles and linear series. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 48. Springer-Verlag, Berlin, 2004
Luna, D.: Grosses cellules pour les varieéteés spheériques. Algebraic groups and Lie groups, 267–280, Austral. Math. Soc. Lect. Ser., 9, Cambridge Univ. Press, Cambridge, 1997
Luna, D., Vust, T.: Plongements d’espaces homogènes. Comment. Math. Helv., 58 (2), 186–245 (1983)
Moser-Jauslin, L.: The Chow rings of smooth complete SL(2)-embeddings. Compositio Math., 82(1), 67–106 (1992)
Pauer, F.: Über gewisse G-stabile Teilmengen in projektiven Räumen. Manusc. Math., 66, 1–16 (1989)
Pasquier, B.: On some smooth projective two-orbit varieties with Picard number 1. Math. Ann., 344 (4), 963–987 (2009)
Pasquier, B., Perrin, N.: Local rigidity of quasi-regular varieties. Math. Z., 265 (3), 589–600 (2010)
Perrin, N.: On the geometry of spherical varieties. Trans. Groups., 19 (1), 171–223 (2014)
Ressayre, N.: Spherical homogeneous spaces of minimal rank. Adv. Math., 224 (5), 1784–1800 (2010)
Richardson, R. W.: On orbits of algebraic groups and Lie groups. Bull. Austral. Math. Soc., 25 (1), 1–28 (1982)
Richardson, R.W., Springer, T. A.: The Bruhat order on symmetric varieties. Geom. Dedicata, 35, 389–436 (1990)
Richardson, R. W., Springer, T. A.: Complements to: “The Bruhat order on symmetric varieties”. Geom. Dedicata, 49 (2), 231–238 (1994)
Rosenlicht, M.: A remark on quotient spaces. An. Acad. Brasil. Ci., 35, 487–489 (1963)
Serre, J. P.: Espaces fibrés algébriques. Séminaire Claude Chevalley, 3 (1958), Exposé No. 1, 37 p
Sumihiro, H.: Equivariant completion. J. Math. Kyoto Univ., 14, 1–28 (1974)
Sumihiro, H.: Equivariant completion. II. J. Math. Kyoto Univ., 15 (3), 573–605 (1975)
Timashëv, D. A.: Homogeneous spaces and equivariant embeddings. Encyclopaedia of Mathematical Sciences, 138. Invariant Theory and Algebraic Transformation Groups, 8. Springer, Heidelberg, 2011
Acknowledgements
I first thank Michel Brion and Baohua Fu for the kind invitation to give lectures on the geometry of spherical varieties in Sanya. I thank Jacopo Gandini for many helpful discussions on our lectures during the two weeks of the conference. I also thank all the participants especially Johannes Hofscheier and Dmitry Timashev for the many questions and discussions during and after the talks. This led to many improvements and expansions of the first version of these notes. Finally I thank the referee for his comments and corrections.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Perrin, N. Sanya Lectures: Geometry of Spherical Varieties. Acta. Math. Sin.-English Ser. 34, 371–416 (2018). https://doi.org/10.1007/s10114-017-7163-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-017-7163-6