## Duke Mathematical Journal

### Flexible varieties and automorphism groups

#### Abstract

Given an irreducible affine algebraic variety $X$ of dimension $n\ge2$, we let $\operatorname {SAut}(X)$ denote the special automorphism group of $X$, that is, the subgroup of the full automorphism group $\operatorname{Aut}(X)$ generated by all one-parameter unipotent subgroups. We show that if $\operatorname {SAut}(X)$ is transitive on the smooth locus $X_{\mathrm{reg}}$, then it is infinitely transitive on $X_{\mathrm{reg}}$. In turn, the transitivity is equivalent to the flexibility of $X$. The latter means that for every smooth point $x\in X_{\mathrm{reg}}$ the tangent space $T_{x}X$ is spanned by the velocity vectors at $x$ of one-parameter unipotent subgroups of $\operatorname{Aut}(X)$. We also provide various modifications and applications.

#### Article information

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

Dates
First available in Project Euclid: 15 March 2013

https://projecteuclid.org/euclid.dmj/1363355693

Digital Object Identifier
doi:10.1215/00127094-2080132

Mathematical Reviews number (MathSciNet)
MR3039680

Zentralblatt MATH identifier
1295.14057

#### Citation

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.

#### References

• [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.