Abstract
In this note we survey recent results on automorphisms of affine algebraic varieties, infinitely transitive group actions and flexibility. We present related constructions and examples, and discuss geometric applications and open problems.
Mathematics Subject Classification codes (2000): 14R20, 14L30
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
By this we mean a nowhere vanishing n-form defined on \(X_{\mathrm{reg}}\).
References
Arzhantsev, I. V., Flenner, H., Kaliman, S., Kutzschebauch, F., and Zaidenberg, M., Flexible varieties and automorphism groups, Duke Math. J., 162 no. 4 (2013), 60p (to appear) arXiv:1011.5375, (2010).
Arzhantsev, I.V., Kuyumzhiyan, K., and Zaidenberg, M., Flag varieties, toric varieties, and suspensions: three instances of infinite transitivity, Sbornik: Math., 203, no. 7, 3–30, (2012).
Batyrev, V. and Haddad, F., On the geometry ofSL (2)-equivariant flips, Moscow Math. J. 8, no. 4, 621–646, (2012).
Bogomolov, F., Karzhemanov, I., and Kuyumzhiyan, K., Unirationality and existence of infinitely transitive models, this volume; see also arXiv:1204.0862, (2012).
Borel, A., Les bouts des espaces homogènes de groupes de Lie, Ann. Math. (2) 58, 443–457, (1953).
Danilov, V. I. and Gizatullin, M. H., Examples of nonhomogeneous quasihomogeneous surfaces, Math. USSR Izv. 8, 43–60, (1974).
Dubouloz, A., Completions of normal affine surfaces with a trivial Makar-Limanov invariant, Michigan Math. J. 52, 289–308, (2004).
Dubouloz, A., The cylinder over the Koras-Russell cubic threefold has a trivial Makar-Limanov invariant, Transform. Groups 14, 531–539, (2009).
Flenner, H., Kaliman, S., and Zaidenberg, M., Smooth Affine Surfaces with Non-Unique \({\mathbb{C}}^{{\ast}}- Actions,\) J. Algebraic Geometry 20, 329–398, (2011).
Freudenburg, G., Algebraic Theory of Locally Nilpotent Derivations, Encyclopaedia Math. Sciences 136, Springer, Berlin, (2006).
Gizatullin, M. H., Affine surfaces that can be augmented by a nonsingular rational curve (Russian), Izv. Akad. Nauk SSSR, Ser. Mat. 34, 778–802, (1970).
Gizatullin, M. H., Quasihomogeneous affine surfaces, Math. USSR Izv. 5, 1057–1081, (1971).
Kaliman, S. and Kutzschebauch, F., Criteria for the density property of complex manifolds, Invent. Math. 172, 71–87, (2008).
Kaliman, S. and Zaidenberg, M., Affine modifications and affine hypersurfaces with a very transitive automorphism group, Transform. Groups 4, 53–95, (1999).
Kaliman, S. and Zaidenberg, M., Miyanishi’s characterization of the affine 3-space does not hold in higher dimensions, Ann. Inst. Fourier (Grenoble) 50, 1649–1669, (2000).
Kishimoto, T., Prokhorov, Yu., and Zaidenberg, M., Group actions on affine cones, CRM Proceedings and Lecture Notes 54, Amer. Math. Soc., 123–164, (2011).
Kishimoto, T., Prokhorov, Yu., and Zaidenberg, M., Unipotent group actions on del Pezzo cones, arXiv:1212.4479, (2012), 10p.
Kishimoto, T., Yu. Prokhorov, and Zaidenberg, M. G a -actions on affine cones. arXiv:1212.4249, (2012), 14p.
Knop, F., Mehrfach transitive Operationen algebraischer Gruppen, Arch. Math. 41, 438–446, (1983).
Kuyumzhiyan, K. and Mangolte, F., Infinitely transitive actions on real affine suspensions, J. Pure Appl. Algebra 216, no. 10, 2106–2112, (2012).
Liendo, A., Affine T-varieties of complexity one and locally nilpotent derivations, Transform. Groups 15, 389–425, (2010).
Liendo, A., \(\mathbb{G}_{a}\)-actions of fiber type on affineT-varieties, J. Algebra 324, 3653–3665, (2010).
Perepechko, A. Yu., Flexibility of a ne cones over del Pezzo surfaces of degree 4 and 5, Funct. Anal. Appl. (to appear); arXiv:1108.5841, 6p.
Popov, V. L., Quasihomogeneous affine algebraic varieties of the groupSL (2), Math. USSR Izv. 7, 793–831, (1973).
Popov, V. L., On the Makar-Limanov, Derksen invariants, and finite automorphism groups of algebraic varieties, CRM Proceedings and Lecture Notes 54, Amer. Math. Soc., 289–311, (2011).
Popov, V. L. and Vinberg, E. B., On a certain class of quasihomogeneous affine varieties, Math. USSR Izv. 6, 743–758, (1972).
Ramanujam, C. P., A note on automorphism groups of algebraic varieties, Math. Ann. 156, 25–33, (1964).
Reichstein, Z., On automorphisms of matrix invariants, Trans. Amer. Math. Soc. 340, 353–371, (1993).
Reichstein, Z., On automorphisms of matrix invariants induced from the trace ring, Linear Algebra Appl. 193, 51–74, (1993).
Shafarevich, I. R., On some infinite-dimensional groups, Rend. Mat. Appl. (5) 25, no. 1–2, 208–212, (1966).
Shestakov, I.P. and Umirbaev, U. U., The tame and the wild automorphisms of polynomial rings in three variables, J. Amer. Math. Soc. 17, 197–227, (2004).
Winkelmann, J., On automorphisms of complements of analytic subsets in \({\mathbb{C}}^{n}\), Math. Z. 204, 117–127, (1990).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this chapter
Cite this chapter
Arzhantsev, I., Flenner, H., Kaliman, S., Kutzschebauch, F., Zaidenberg, M. (2013). Infinite Transitivity on Affine Varieties. In: Bogomolov, F., Hassett, B., Tschinkel, Y. (eds) Birational Geometry, Rational Curves, and Arithmetic. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6482-2_1
Download citation
DOI: https://doi.org/10.1007/978-1-4614-6482-2_1
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-6481-5
Online ISBN: 978-1-4614-6482-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)