Automorphisms of algebraic varieties and infinite transitivity

I. Arzhantsev

Geodesic

Parcourir par

Automorphisms of algebraic varieties and infinite transitivity

I. Arzhantsev

Algebra i analiz, Tome 34 (2022) no. 2, pp. 1-55

Voir la notice de l'article provenant de la source Math-Net.Ru

Résumé

The paper reviews the results of recent years on the multiple transitivity of actions of the automorphism groups of affine algebraic varieties. The property of infinite transitivity for the action of the group of special automorphisms is considered and the equivalent flexibility property of the variety. These properties have important algebraic and geometric consequences, and at the same time they are fulfilled for wide classes of manifolds. The cases when infinite transitivity occurs for automorphism groups generated by a finite number of one-parameter subgroups are studied separately. In the appendices to the paper, the results on infinitely transitive actions in complex analysis and in combinatorial group theory are considered.

Keywords: algebraic manifold, multiple transitivity, locally nilpotent differentiation, toric manifold, affine cone.
Mots-clés : automorphism, group action

@article{AA_2022_34_2_a0,
     author = {I. Arzhantsev},
     title = {Automorphisms of algebraic varieties and infinite transitivity},
     journal = {Algebra i analiz},
     pages = {1--55},
     publisher = {mathdoc},
     volume = {34},
     number = {2},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AA_2022_34_2_a0/}
}

TY  - JOUR
AU  - I. Arzhantsev
TI  - Automorphisms of algebraic varieties and infinite transitivity
JO  - Algebra i analiz
PY  - 2022
SP  - 1
EP  - 55
VL  - 34
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AA_2022_34_2_a0/
LA  - ru
ID  - AA_2022_34_2_a0
ER  -

%0 Journal Article
%A I. Arzhantsev
%T Automorphisms of algebraic varieties and infinite transitivity
%J Algebra i analiz
%D 2022
%P 1-55
%V 34
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AA_2022_34_2_a0/
%G ru
%F AA_2022_34_2_a0

I. Arzhantsev. Automorphisms of algebraic varieties and infinite transitivity. Algebra i analiz, Tome 34 (2022) no. 2, pp. 1-55. http://geodesic.mathdoc.fr/item/AA_2022_34_2_a0/

Bibliographie
Cité par

[1] Arzhantsev I. V., Zaidenberg M. G., Kuyumzhiyan K. G., “Mnogoobraziya flagov, toricheskie mnogoobraziya i nadstroiki: tri primera beskonechnoi tranzitivnosti”, Mat. sb., 203:7 (2012), 3–30 | MR | Zbl | MR | Zbl

[2] Bodnarchuk Yu. V., “Pro tranzitivnist diï grupi biregulyarnikh avtomorfizmiv afinnikh prostoriv, scho mistyat afinnu pidgrupu”, Dopov. Akad. Nauk Ukraïni, 1995, no. 1, 5–7

[3] Boldyrev I. A., Gaifullin S. A., “Avtomorfizmy nenormalnykh toricheskikh mnogoobrazii”, Mat. zametki, 110:6 (2021), 837–855 | MR | Zbl | MR | Zbl

[4] Vinberg E. B., Popov V. L., “Ob odnom klasse kvaziodnorodnykh affinnykh mnogoobrazii”, Izv. AN SSSR. Ser. mat., 36:4 (1972), 749–764 | Zbl | Zbl

[5] Vinberg E. B., Popov V. L., “Teoriya invariantov”, Algebraicheskaya geometriya. 4, Itogi nauki i tekhn. Sovrem. probl. mat. Fundam. napravleniya, 55, VINITI, M., 1989, 137–309

[6] Gizatullin M. Kh., “Ob affinnykh poverkhnostyakh, popolnyaemykh neosoboi ratsionalnoi krivoi”, Izv. AN SSSR. Ser. mat., 34:4 (1970), 778–802 | MR | Zbl | MR | Zbl

[7] Gizatullin M. Kh., “Kvaziodnorodnye affinnye poverkhnosti”, Izv. AN SSSR. Ser. mat., 35:5 (1971), 1047–1071 | MR | Zbl | MR | Zbl

[8] Gizatullin M. Kh., Danilov V. I., “Primery neodnorodnykh kvaziodnorodnykh poverkhnostei”, Izv. AN SSSR. Ser. mat., 38:1 (1974), 42–58 | Zbl | Zbl

[9] Perepechko A. Yu., “Gibkost affinnykh konusov nad poverkhnostyami del Petstso stepenei $4$ i $5$”, Funktsional. anal. i ego pril., 47:4 (2013), 45–52 | MR | Zbl | MR | Zbl

[10] Popov V. L., “Kvaziodnorodnye affinnye algebraicheskie mnogoobraziya gruppy $\mathrm{SL}(2)$”, Izv. AN SSSR. Ser. mat., 37:4 (1973), 792–832 | Zbl | Zbl

[11] Shafarevich A. A., “Gibkost $S$-mnogoobrazii poluprostykh grupp”, Mat. cb., 208:2 (2017), 121–148 | Zbl | Zbl

[12] Andrist R. B., “Integrable generators of Lie algebras of vector fields on $\mathbb{C}^n$”, Forum Math., 31:4 (2019), 943–49 | DOI | MR | DOI | MR

[13] Andrist R. B., “The density property for Calogero–Moser spaces”, Proc. Amer. Math. Soc., 149:10 (2021), 4207–4218 | DOI | MR | Zbl | DOI | MR | Zbl

[14] Andrist R. B., Ugolini R., “A new notion of tameness”, J. Math. Analysis Appl., 472:1 (2019), 196–215 | DOI | MR | Zbl | DOI | MR | Zbl

[15] Arzhantsev I., “Infinite transitivity and special automorphisms”, Ark. Mat., 56:1 (2018), 1–14 | DOI | MR | Zbl | DOI | MR | Zbl

[16] Arzhantsev I., Derenthal U., Hausen J., Laface A., Cox rings, Cambridge Stud. Adv. Math., 144, Cambridge Univ. Press, Cambridge, 2015 | MR | Zbl | MR | Zbl

[17] Arzhantsev I., Flenner H., Kaliman Sh., Kutzschebauch F., Zaidenberg M., “Flexible varieties and automorphism groups”, Duke Math. J., 162:4 (2013), 767–823 | DOI | MR | Zbl | DOI | MR | Zbl

[18] Arzhantsev I., Flenner H., Kaliman S., Kutzschebauch F., Zaidenberg M., “Infinite transitivity on affine varieties”, Birational geometry, rational curves, and arithmetic, Simons Symp., Springer, Cham, 2013, 1–13 | MR | Zbl | MR | Zbl

[19] Arzhantsev I., Gaifullin S., “The automorphism group of a rigid affine variety”, Math. Nachr., 290:5-6 (2017), 662–671 | DOI | MR | Zbl | DOI | MR | Zbl

[20] Arzhantsev I., Kuyumzhiyan K., Zaidenberg M., “Infinite transitivity, finite generation, and Demazure roots”, Adv. Math., 351 (2019), 1–32 | DOI | MR | Zbl | DOI | MR | Zbl

[21] Arzhantsev I., Liendo A., Stasyuk T., “Lie algebras of vertical derivations on semiaffine varieties with torus actions”, J. Pure Appl. Algebra, 225:2 (2021), 106499 | DOI | MR | Zbl | DOI | MR | Zbl

[22] Arzhantsev I., Perepechko A., Suess H., “Infinite transitivity on universal torsors”, J. London Math. Soc. (2), 89:3 (2014), 762–778 | DOI | MR | Zbl | DOI | MR | Zbl

[23] Arzhantsev I., Zaidenberg M., “Tits alternative and highly transitive actions on toric varieties”, Int. Math. Res. Not. IMRN, 2021, 34 pp. | DOI | DOI

[24] Batyrev V., Haddad F., “On the geometry of ${\rm SL}(2)$-equivariant flips”, Moscow Math. J., 2012, no. 4, 621–646 | DOI | MR | DOI | MR

[25] Berest Yu., Eshmatov A., Eshmatov F., “Multitransitivity of Calogero–Moser spaces”, Transform. Groups, 21:1 (2016), 35–50 | DOI | MR | Zbl | DOI | MR | Zbl

[26] Bogomolov F., Karzhemanov I., Kuyumzhiyan K., “Unirationality and existence of infinitely transitive models”, Birational geometry, rational curves, and arithmetic, Simons Symp., Springer, Cham, 2013, 77–92 | MR | Zbl | MR | Zbl

[27] Borel A., “Les bouts des espaces homogènes de groupes de Lie”, Ann. Math. (2), 58 (1953), 443–457 | DOI | MR | Zbl | DOI | MR | Zbl

[28] Chaynikov V. V., Properties of hyperbolic groups: free normal subgroups, quasiconvex subgroups and actions of maximal growth, Ph.D. Thesis, Vanderbilt Univ., 2012 | MR | MR

[29] Cheltsov I., Park J., Prokhorov Yu., Zaidenberg M., “Cylinders in Fano varieties”, EMS Surv. Math. Sci., 8:1-2 (2021), 39–105 | DOI | MR | Zbl | DOI | MR | Zbl

[30] Cheltsov I., Park J., Won J., “Cylinders in singular del Pezzo surfaces”, Compos. Math., 152:6 (2016), 1198–1224 | DOI | MR | Zbl | DOI | MR | Zbl

[31] Cheltsov I., Park J., Won J., “Affine cones over smooth cubic surfaces”, J. Eur. Math. Soc. (JEMS), 18:7 (2016), 1537–1564 | DOI | MR | Zbl | DOI | MR | Zbl

[32] Chistopolskaya A., “On nilpotent generators of the special linear Lie algebra”, Linear Algebra Appl., 559 (2018), 73–79 | DOI | MR | Zbl | DOI | MR | Zbl

[33] Colliot-Thélène J.-L., Sansuc J.-J., “Torseurs sous des groupes de type multiplicatif; applications à l'étude des points rationnels de certaines variétés algébriques”, C. R. Acad. Sci. Paris Sér. A-B, 282:18 (1976), A1113–A1116 | MR | MR

[34] Colliot-Thélène J.-L., Sansuc J.-J., “Variétés de premiére descente attachées aux variétés rationnelles”, C. R. Acad. Sci. Paris Sér. A-B, 284:16 (1977), A967–A970 | MR | MR

[35] Cox D. A., “The homogeneous coordinate ring of a toric variety”, J. Algebraic Geom., 4 (1995), 17–50 | MR | Zbl | MR | Zbl

[36] Demazure M., “Sous-groupes algebriques de rang maximum du groupe de Cremona”, Ann. Sci. École Norm. Sup. (4), 3 (1970), 507–588 | DOI | MR | Zbl | DOI | MR | Zbl

[37] Dixon J. D., “Most finitely generated permutation groups are free”, Bull. London Math. Soc., 22:3 (1990), 222–226 | DOI | MR | Zbl | DOI | MR | Zbl

[38] Dixon J. D., Mortimer B., Permutation groups, Grad. Texts in Math., 163, Springer-Verlag, New York, 1996 | DOI | MR | Zbl | DOI | MR | Zbl

[39] Dubouloz A., “Completions of normal affine surfaces with a trivial Makar–Limanov invariant”, Michigan Math. J., 52:2 (2004), 289–308 | DOI | MR | Zbl | DOI | MR | Zbl

[40] Dubouloz A., “Flexible bundles over rigid affine surfaces”, Comment. Math. Helv., 90:1 (2015), 121–137 | DOI | MR | Zbl | DOI | MR | Zbl

[41] Flenner H., Kaliman S., Zaidenberg M., “The Gromov–Winkelmann theorem for flexible varieties”, J. Eur. Math. Soc. (JEMS), 18:11 (2016), 2483–2510 | DOI | MR | Zbl | DOI | MR | Zbl

[42] Fima P., Le Maitre F., Moon S., Stalder Y., A characterization of high transitivity for groups acting on trees, arXiv: 2003.11116

[43] Fima P., Moon S., Stalder Y., “Highly transitive actions of groups acting on trees”, Proc. Amer. Math. Soc., 143:12 (2015), 5083–5095 | DOI | MR | Zbl | DOI | MR | Zbl

[44] Forstneric F., Developments in Oka theory since, 2017, arXiv: 2006.07888

[45] Freudenburg G., Algebraic theory of locally nilpotent derivations, Encycl. Math. Sci., 136, Springer-Verlag, Berlin, 2006 | MR | Zbl | MR | Zbl

[46] Furter J.-P., Kraft H., On the geometry of the automorphism group of affine $n$-space, arXiv: 1809.04175

[47] Gaifullin S., On rigidity of trinomial hypersurfaces and factorial trinomial varieties, arXiv: 1902.06136

[48] Gaifullin S., Shafarevich I., “Flexibility of normal affine horospherical varieties”, Proc. Amer. Math. Soc., 147:8 (2019), 3317–3330 | DOI | MR | Zbl | DOI | MR | Zbl

[49] Glass A. M. W., McCleary S. H., “Highly transitive representations of free groups and free products”, Bull. Austral. Math. Soc., 43 (1991), 19–36 | DOI | MR | DOI | MR

[50] Garion S., Glasner Y., “Highly transitive actions of $\text{Out}(F_n)$”, Groups Geom. Dyn., 7:2 (2013), 357–376 | DOI | MR | Zbl | DOI | MR | Zbl

[51] Gromov M., “Oka's principle for holomorphic sections of elliptic bundles”, J. Amer. Math. Soc., 2:4 (1989), 851–897 | MR | Zbl | MR | Zbl

[52] Gunhouse S. V., “Highly transitive representations of free products on the natural numbers”, Arch. Math. (Basel), 58:5 (1992), 435–443 | DOI | MR | Zbl | DOI | MR | Zbl

[53] Hickin K. K., “Highly transitive Jordan representations of free products”, J. London Math. Soc. (2), 46:1 (1992), 81–91 | DOI | MR | Zbl | DOI | MR | Zbl

[54] Hull M., Osin D., “Transitivity degrees of countable groups and acylindrical hyperbolicity”, Israel J. Math., 216:1 (2016), 307–353 | DOI | MR | Zbl | DOI | MR | Zbl

[55] Kaliman S., “Extensions of isomorphisms of subvarieties in flexible varieties”, Transform. Groups, 25:2 (2020), 517–575 | DOI | MR | Zbl | DOI | MR | Zbl

[56] Kaliman S., Kutzschebauch F., “On the present state of the Andersen–Lempert theory”, Affine algebraic geometry, CRM Proc. Lecture Notes, 54, Amer. Math. Soc., Providence, RI, 2011, 85–122 | DOI | MR | Zbl | DOI | MR | Zbl

[57] Kaliman S., Kutzschebauch F., Truong T. T., “On subelliptic manifolds”, Israel J. Math., 228:1 (2018), 229–247 | DOI | MR | Zbl | DOI | MR | Zbl

[58] Kaliman S., Udumyan D., “On automorphisms of flexible varieties”, Adv. Math., 396 (2022), 108112 | DOI | MR | Zbl | DOI | MR | Zbl

[59] Kaliman S., Zaidenberg M., “Affine modifications and affine hypersurfaces with a very transitive automorphism group”, Transform. Groups, 4:1 (1999), 53–95 | DOI | MR | Zbl | DOI | MR | Zbl

[60] Kishimoto T., Prokhorov Yu., Zaidenberg M., “Unipotent group actions on del Pezzo cones”, Algebr. Geom., 1 (2014), 46–56 | DOI | MR | Zbl | DOI | MR | Zbl

[61] Kitroser D., “Highly transitive actions of surface groups”, Proc. Amer. Math. Soc., 140:10 (2012), 3365–3375 | DOI | MR | Zbl | DOI | MR | Zbl

[62] Knop F., “Mehrfach transitive Operationen algebraischer Gruppen”, Arch. Math. (Basel), 41:5 (1983), 438–446 | DOI | MR | Zbl | DOI | MR | Zbl

[63] Kovalenko S., “Transitivity of automorphism groups of Gizatullin surfaces”, Int. Math. Res. Not. IMRN, 21 (2015), 11433–11484 | DOI | MR | Zbl | DOI | MR | Zbl

[64] Kovalenko S., Perepechko A., Zaidenberg M., “On automorphism groups of affine suraces”, Algebraic varieties and automorphism groups, Adv. Stud. Pure Math., 75, Math. Soc. Japan, Tokyo, 2017, 207–286 | DOI | MR | Zbl | DOI | MR | Zbl

[65] Kramer L., “Two-transitive Lie groups”, J. Reine Angew. Math., 563 (2003), 83–113 | MR | Zbl | MR | Zbl

[66] Kutzschebauch F., “Flexibility properties in complex analysis and affine algebraic geometry”, Automorphisms in birational and affine geometry, Springer Proc. Math. Stat., 79, Springer, Cham, 2014, 387–406 | MR | MR

[67] Kuyumzhiyan K., “Infinite transitivity for Calogero–Moser spaces”, Proc. Amer. Math. Soc., 148:9 (2020), 3723–3731 | DOI | MR | Zbl | DOI | MR | Zbl

[68] Kuyumzhiyan K., Mangolte F., “Infinitely transitive actions on real affine suspensions”, J. Pure Appl. Algebra, 216:10 (2012), 2106–2112 | DOI | MR | Zbl | DOI | MR | Zbl

[69] Lárusson F., Truong T. T., “Approximation and interpolation of regular maps from affine varieties to algebraic manifolds”, Math. Scand., 125:2 (2019), 199–209 | DOI | MR | Zbl | DOI | MR | Zbl

[70] Lewis D., Perry K., Straub A., “An algorithmic approach to the polydegree conjecture for plane polynomial automorphisms”, J. Pure Appl. Algebra, 223:12 (2019), 5346–5359 | DOI | MR | Zbl | DOI | MR | Zbl

[71] Liendo A., “Affine $T$-varieties of complexity one and locally nilpotent derivations”, Transform. Groups, 15:2 (2010), 389–425 | DOI | MR | Zbl | DOI | MR | Zbl

[72] Liendo A., “$\mathbb{G}_a$-actions of fiber type on affine $T$-varieties”, J. Algebra, 324:12 (2010), 3653–3665 | DOI | MR | Zbl | DOI | MR | Zbl

[73] Matsusaka T., “Polarized varieties, fields of moduli and generalized Kummer varieties of polarized abelian varieties”, Amer. J. Math., 80 (1958), 45–82 | DOI | MR | Zbl | DOI | MR | Zbl

[74] McDonough T. P., “A permutation representation of a free group”, Quart. J. Math. Oxford Ser. (2), 28:111 (1977), 353–356 | DOI | MR | Zbl | DOI | MR | Zbl

[75] Michałek M., Perepechko A., Suess H., “Flexible affine cones and flexible coverings”, Math. Z., 290:3-4 (2018), 1457–1478 | DOI | MR | Zbl | DOI | MR | Zbl

[76] Moon S., Stalder Y., “Highly transitive actions of free products”, Algebr. Geom. Topol., 13:1 (2013), 589–607 | DOI | MR | Zbl | DOI | MR | Zbl

[77] Park J., Won J., “Flexible affine cones over del Pezzo surfaces of degree $4$”, Eur. J. Math., 2 (2016), 304–318 | DOI | MR | Zbl | DOI | MR | Zbl

[78] Perepechko A., “Affine cones over cubic surfaces are flexible in codimension one”, Forum Math., 33:2 (2021), 339–348 | DOI | MR | DOI | MR

[79] Popov V., “On the Makar–Limanov, Derksen invariants, and finite automorphism groups of algebraic varieties”, CRM Proc. Lecture Notes, 54, Amer. Math. Soc., Providence, RI, 2011, 289–311 | DOI | MR | Zbl | DOI | MR | Zbl

[80] Popov V., “On infinite dimensional algebraic transformation groups”, Transform. Groups, 19:2 (2014), 549–568 | DOI | MR | Zbl | DOI | MR | Zbl

[81] Prokhorov Yu., Zaidenberg M., Affine cones over Fano–Mukai fourfolds of genus $10$ are flexible, arXiv: 2005.12092

[82] Reichstein Z., “On automorphisms of matrix invariants”, Trans. Amer. Math. Soc., 340 (1993), 353–371 | DOI | MR | Zbl | DOI | MR | Zbl

[83] Reichstein Z., “On automorphisms of matrix invariants induced from the trace ring”, Linear Algebra Appl., 193 (1993), 51–74 | DOI | MR | Zbl | DOI | MR | Zbl

[84] Rosay J.-P., Rudin W., “Holomorphic maps from $\mathbb{C}^n$ to $\mathbb{C}^n$”, Trans. Amer. Math. Soc., 310:1 (1988), 47–86 | MR | Zbl | MR | Zbl

[85] Tits J., “Free subgroups in linear groups”, J. Algebra, 20 (1972), 250–270 | DOI | MR | Zbl | DOI | MR | Zbl

[86] Internat. J. Math., 11:6 (2000), 837–847 | DOI | DOI | MR | MR | Zbl | Zbl | DOI | MR | Zbl

[87] Winkelmann J., “On automorphisms of complements of analytic subsets in $\mathbb{C}^n$”, Math. Z., 204 (1990), 117–127 | DOI | MR | Zbl | DOI | MR | Zbl

[88] Winkelmann J., “Tameness and growth conditions”, Doc. Math., 13 (2008), 97–101 | MR | Zbl | MR | Zbl

[89] Winkelmann J., “Tame discrete subsets in Stein manifolds”, J. Aust. Math. Soc., 107:1 (2019), 110–132 | DOI | MR | Zbl | DOI | MR | Zbl