Duke Mathematical Journal

Flexible varieties and automorphism groups

I. Arzhantsev, H. Flenner, S. Kaliman, F. Kutzschebauch, and M. Zaidenberg

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Given an irreducible affine algebraic variety X of dimension n2, we let SAut(X) denote the special automorphism group of X, that is, the subgroup of the full automorphism group Aut(X) generated by all one-parameter unipotent subgroups. We show that if SAut(X) is transitive on the smooth locus Xreg, then it is infinitely transitive on Xreg. In turn, the transitivity is equivalent to the flexibility of X. The latter means that for every smooth point xXreg the tangent space TxX is spanned by the velocity vectors at x of one-parameter unipotent subgroups of Aut(X). We also provide various modifications and applications.

Article information

Duke Math. J., Volume 162, Number 4 (2013), 767-823.

First available in Project Euclid: 15 March 2013

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Zentralblatt MATH identifier

Primary: 14R20: Group actions on affine varieties [See also 13A50, 14L30] 32M17: Automorphism groups of Cn and affine manifolds
Secondary: 14L30: Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17]


Arzhantsev, I.; Flenner, H.; Kaliman, S.; Kutzschebauch, F.; Zaidenberg, M. Flexible varieties and automorphism groups. Duke Math. J. 162 (2013), no. 4, 767--823. doi:10.1215/00127094-2080132. https://projecteuclid.org/euclid.dmj/1363355693

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  • [1] I. V. Arzhantsev, M. G. Zaidenberg, and K. G. Kuyumzhiyan, Flag varieties, toric varieties, and suspensions: Three instances of infinite transitivity, Sb. Math. 203 (2012), 923–949.
  • [2] V. Batyrev and F. Haddad, On the geometry of $\operatorname{SL}(2)$-equivariant flips, Mosc. Math. J. 8 (2008), 621–646, 846.
  • [3] A. Białynicki-Birula, G. Hochschild, and G. D. Mostow, Extensions of representations of algebraic linear groups, Amer. J. Math. 85 (1963), 131–144.
  • [4] F. Bogomolov, I. Karzhemanov, and K. Kuyumzhiyan, “Unirationality and existence of infinitely transitive models,” to appear in Birational Geometry, Rational Curves, and Arithmetic, preprint, arXiv:1204.0862v3 [math.AG].
  • [5] A. Borel, Les bouts des espaces homogènes de groupes de Lie, Ann. of Math. (2) 58 (1953), 443–457.
  • [6] G. T. Buzzard and F. Forstneric, An interpolation theorem for holomorphic automorphisms of $\mathbf{C}^n$, J. Geom. Anal. 10 (2000), 101–108.
  • [7] V. I. Danilov, “Algebraic varieties and schemes” in Algebraic Geometry, I, Encyclopaedia Math. Sci. 23, Springer, Berlin, 1994, 167–297.
  • [8] F. Donzelli, Algebraic density property of Danilov–Gizatullin surfaces, Math. Z. 272 (2012), 1187–1194.
  • [9] F. Donzelli, Makar-Limanov invariant, Derksen invariant, flexible points, preprint, arXiv:1107.3340v1 [math.AG].
  • [10] A. Dubouloz, Completions of normal affine surfaces with a trivial Makar-Limanov invariant, Michigan Math. J. 52 (2004), 289–308.
  • [11] H. Flenner, S. Kaliman, and M. Zaidenberg, Smooth affine surfaces with nonunique $\mathbb{C}^*$-actions, J. Algebraic Geom. 20 (2011), 329–398.
  • [12] F. Forstnerič, “The homotopy principle in complex analysis: a survey” in Explorations in Complex and Riemannian Geometry, Contemp. Math. 332, Amer. Math. Soc., Providence, 2003, 73–99.
  • [13] F. Forstnerič, Holomorphic flexibility properties of complex manifolds, Amer. J. Math. 128 (2006), 239–270.
  • [14] F. Forstnerič, Stein manifolds and holomorphic mappings, Ergeb. Math. Grenzgeb. (3) 56, Springer, Berlin, 2011.
  • [15] G. Freudenburg, Algebraic theory of locally nilpotent derivations, Encyclopaedia Math. Sci. 136, Springer, Berlin, 2006.
  • [16] G. Freudenburg and P. Russell, “Open problems in affine algebraic geometry: 11, Problems by V. Popov” in Affine Algebraic Geometry, Contemp. Math. 369, Amer. Math. Soc., Providence, 2005, 1–30.
  • [17] J.-P. Furter and S. Lamy, Normal subgroup generated by a plane polynomial automorphism, Transform. Groups 15 (2010), 577–610.
  • [18] S. A. Gaifullin, Affine toric $\operatorname{SL}(2)$-embeddings, Sb. Math. 199 (2008), 319–339.
  • [19] M. H. Gizatullin, Affine surfaces that can be augmented by a nonsingular rational curve, Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), 778–802.
  • [20] M. H. Gizatullin, Affine surfaces that are quasihomogeneous with respect to an algebraic group, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 738–753.
  • [21] M. H. Gizatullin, Quasihomogeneous affine surfaces, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 1047–1071.
  • [22] M. H. Gizatullin and V. I. Danilov, Examples of nonhomogeneous quasihomogeneous surfaces, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 42–58.
  • [23] M. Gromov, Oka’s principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2 (1989), 851–897.
  • [24] A. Grothendieck, Éléments de géométrie algébrique, II: Étude globale élémentaire de quelques classes de morphismes, Inst. Hautes Études Sci. Publ. Math. 8 (1961). IV: Étude locale des schémas et des morphismes de schémas III, 28 (1966).
  • [25] R. Hartshorne, Algebraic geometry, Grad. Texts in Math. 52, Springer, New York, 1977.
  • [26] Z. Jelonek, A hypersurface which has the Abhyankar-Moh property, Math. Ann. 308 (1997), 73–84.
  • [27] S. Kaliman and F. Kutzschebauch, Criteria for the density property of complex manifolds, Invent. Math. 172 (2008), 71–87.
  • [28] S. Kaliman and F. Kutzschebauch, Algebraic volume density property of affine algebraic manifolds, Invent. Math. 181 (2010), 605–647.
  • [29] S. Kaliman and F. Kutzschebauch, “On the present state of the Andersen-Lempert theory” in Affine Algebraic Geometry, CRM Proc. Lecture Notes 54, Amer. Math. Soc., Providence, 2011, 85–122.
  • [30] S. Kaliman and M. Zaidenberg, Affine modifications and affine hypersurfaces with a very transitive automorphism group, Transform. Groups 4 (1999), 53–95.
  • [31] S. Kaliman, M. Zaidenberg, Miyanishi’s characterization of the affine 3-space does not hold in higher dimensions, Ann. Inst. Fourier (Grenoble) 50 (2000), 1649–1669.
  • [32] S. L. Kleiman, The transversality of a general translate, Compos. Math. 28 (1974), 287–297.
  • [33] F. Knop, Mehrfach transitive Operationen algebraischer Gruppen, Arch. Math. (Basel) 41 (1983), 438–446.
  • [34] H. Kraft, Geometrische Methoden in der Invariantentheorie, Aspects Math. D1, Vieweg, Braunschweig, 1984.
  • [35] S. Kumar, Kac-Moody groups, their flag varieties, and representation theory, Progr. Math. 204, Birkhäuser, Boston, 2002.
  • [36] A. Liendo, Affine $\mathbb{T}$-varieties of complexity one and locally nilpotent derivations, Transform. Groups 15 (2010), 389–425.
  • [37] A. Liendo, $\mathbb{G}_a$-actions of fiber type on affine $\mathbb{T}$-varieties, J. Algebra 324 (2010), 3653–3665.
  • [38] L. Makar-Limanov, On groups of automorphisms of a class of surfaces, Israel J. Math. 69 (1990), 250–256.
  • [39] L. Makar-Limanov, “Locally nilpotent derivations on the surface $xy=p(z)$” in Proceedings of the Third International Algebra Conference (Tainan, 2002), Kluwer, Dordrecht, 2003, 215–219.
  • [40] A. Perepechko, Flexibility of affine cones over del Pezzo surfaces of degree 4 and 5, to appear in Funct. Anal. Appl., preprint, arXiv:1108.5841v1 [math.AG].
  • [41] V. L. Popov, Classification of affine algebraic surfaces that are quasihomogeneous with respect to an algebraic group, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 1038–1055.
  • [42] V. L. Popov, Quasihomogeneous affine algebraic varieties of the group $\operatorname{SL}(2)$, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 792–832.
  • [43] V. L. Popov, Classification of three-dimensional affine algebraic varieties that are quasihomogeneous with respect to an algebraic group, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no. 3, 566–609, 703.
  • [44] V.L. Popov, “On actions of $\mathbf{G}_a$ on $\mathbf{A}^n$” in Algebraic Groups, Utrecht 1986, Lecture Notes in Math. 1271, Springer, Berlin, 1987, 237–242.
  • [45] V.L. Popov, “Generically multiple transitive algebraic group actions” in Algebraic Groups and Homogeneous Spaces, Tata Inst. Fund. Res. Stud. Math., Tata Inst. Fund. Res., Mumbai, 2007, 481–523.
  • [46] V. L. Popov, “On the Makar-Limanov, Derksen invariants, and finite automorphism groups of algebraic varieties” in Affine Algebraic Geometry, CRM Proc. Lecture Notes 54, Amer. Math. Soc., Providence, 2011, 289–312.
  • [47] V. L. Popov and E. B. Vinberg, “Invariant theory” in Algebraic Geometry, IV, Encyclopaedia Math. Sci. 55, Springer, Berlin, 1994, 123–278.
  • [48] C. Procesi, Lie Groups: An Approach through Invariants and Representations, Universitext, Springer, New York, 2007.
  • [49] C. P. Ramanujam, A note on automorphism groups of algebraic varieties, Math. Ann. 156 (1964), 25–33.
  • [50] Z. Reichstein, On automorphisms of matrix invariants, Trans. Amer. Math. Soc. 340 (1993), no. 1, 353–371.
  • [51] Z. Reichstein, On automorphisms of matrix invariants induced from the trace ring, Linear Algebra Appl. 193 (1993), 51–74.
  • [52] R. Rentschler, Opérations du groupe additif sur le plan affine, C. R. Acad. Sci. Paris Sér. A 267 (1968), 384–387.
  • [53] I. R. Shafarevich, On some infinite-dimensional groups, Rend. Mat. e Appl. (5) 25 (1966), 208–212.
  • [54] I. P. Shestakov and U. U. Umirbaev, The tame and the wild automorphisms of polynomial rings in three variables, J. Amer. Math. Soc. 17 (2004), 197–227.
  • [55] D. Varolin, A general notion of shears, and applications, Michigan Math. J. 46 (1999), 533–553.
  • [56] D. Varolin, The density property for complex manifolds and geometric structures, I, J. Geom. Anal. 11 (2001), 135–160. II, Internat. J. Math. 11 (2000), 837–847.
  • [57] J. Winkelmann, On automorphisms of complements of analytic subsets in $\mathbf{C}^n$, Math. Z. 204 (1990), 117–127.