Algebraic overshear density property

Rafael B. Andrist1, Frank Kutzschebauch2
1Department of Mathematics, American University of Beirut, Beirut, Lebanon
2Mathematical Institute, University of Bern, Bern, Switzerland

Tóm tắt

We introduce the notion of the algebraic overshear density property which implies both the algebraic notion of flexibility and the holomorphic notion of the density property. We investigate basic consequences of this stronger property, and propose further research directions in this borderland between affine algebraic geometry and elliptic holomorphic geometry. As an application, we show that any smoothly bordered Riemann surface with finitely many boundary components that is embedded in a complex affine surface with the algebraic overshear density property admits a proper holomorphic embedding.

Tài liệu tham khảo

Alarcón, A., Forstnerič, F., López, F. J.: Minimal surfaces from a complex analytic viewpoint. Springer Monographs in Mathematics, Springer, Cham (2021). https://doi.org/10.1007/978-3-030-69056-4 Alarcón, A., Forstnerič, F.: Embedded complex curves in the affine plane. arXiv:2301.10304 (2023) Andersén, E., Lempert, L.: On the group of holomorphic automorphisms of \({ C}^n\). Invent. Math. 110(2), 371–388 (1992). https://doi.org/10.1007/BF01231337 Andrist, R.B.: Stein spaces characterized by their endomorphisms. Trans. Am. Math. Soc. 363(5), 2341–2355 (2011). https://doi.org/10.1090/S0002-9947-2010-05104-9 Andrist, R.B., Wold, E.F.: Riemann surfaces in Stein manifolds with the density property, language=English, with English and French summaries. Ann. Inst. Fourier (Grenoble) 64(2), 681–697 (2014) Andrist, R.B., Wold, E.F.: Free dense subgroups of holomorphic automorphisms. Math. Z. 280(1–2), 335–346 (2015). https://doi.org/10.1007/s00209-015-1425-8 Andrist, R.B., Kutzschebauch, F.: The fibred density property and the automorphism group of the spectral ball. Math. Ann. 370(1–2), 917–936 (2018). https://doi.org/10.1007/s00208-017-1520-8 Andrist, R.B.: The density property for Gizatullin surfaces with reduced degenerate fibre. J. Geom. Anal. 28(3), 2522–2538 (2018). https://doi.org/10.1007/s12220-017-9916-y Andrist, R.B.: The density property for Calogero-Moser spaces. Proc. Amer. Math. Soc. 149(10), 4207–4218 (2021). https://doi.org/10.1090/proc/15457 Arzhantsev, I., Flenner, H., Kaliman, S., Kutzschebauch, F., Zaidenberg, M.: Flexible varieties and automorphism groups. Duke Math. J. 162(4), 767–823 (2013). https://doi.org/10.1215/00127094-2080132 Bandman, T., Makar-Limanov, L.: Affine surfaces with \({\rm AK}(S)=\mathbb{C} \). Michigan Math. J. 49(3), 567–582 (2001). https://doi.org/10.1307/mmj/1012409971 Buzzard, G.T., Forstnerič, F.: A Carleman type theorem for proper holomorphic embeddings. Ark. Mat. 35(1), 157–169 (1997) Derksen, H., Kutzschebauch, F.: Nonlinearizable holomorphic group actions. Math. Ann. 311(1), 41–53 (1998) Donzelli, F., Dvorsky, A., Kaliman, S.: Algebraic density property of homogeneous spaces. Transform. Groups 15(3), 551–576 (2010). https://doi.org/10.1007/s00031-010-9091-8 Dubouloz, A.: Completions of normal affine surfaces with a trivial Makar-Limanov invariant. Michigan Math. J. 52(2), 289–308 (2004). https://doi.org/10.1307/mmj/1091112077 Fornæss, J.E., Forstnerič, F., Wold, E.F.: Holomorphic approximation: the legacy of Weierstrass, Runge, Oka-Weil, and Mergelyan, 133–192 (2020). https://doi.org/10.1007/978-3-030-40120-7_5 Forstnerič, F., Rosay, J.P.: Approximation of biholomorphic mappings by automorphisms of \({ C}^n\). Invent. Math. 112(2), 323–349 (1993a). https://doi.org/10.1007/BF01232438 Forstnerič, F., Rosay, J.-P.: Erratum: “Approximation of biholomorphic mappings by automorphisms of \({\bf C}^n\)” [Invent. Math. 112 (1993), no. 2, 323–349; MR1213106 (94f:32032)], Invent. Math. 118(3), 573–574 (1993b). https://doi.org/10.1007/BF01231544 Forstnerič, F., Wold, E.F.: Bordered Riemann surfaces in \(\mathbb{C}^2\), English, with English and French summaries. J. Math. Pures Appl. 91(1), 100–114 (2009). https://doi.org/10.1016/j.matpur.2008.09.010 Forstnerič, F.: Mergelyan’s and Arakelian’s theorems for manifold-valued maps. Mosc. Math. J. 19(3), 465–484 (2019). https://doi.org/10.17323/1609-4514-2019-19-3-465-484 Forstnerič, F.: Stein manifolds and holomorphic mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 56, 2, The homotopy principle in complex analysis, Springer, Cham (2017). https://doi.org/10.1007/978-3-319-61058-0 Freudenburg, G.: Algebraic theory of locally nilpotent derivations. Encyclopaedia of Mathematical Sciences, 136, 2, Invariant Theory and Algebraic Transformation Groups, VII, Springer-Verlag. Berlin (2017). https://doi.org/10.1007/978-3-662-55350-3 Kanel-Belov, A., Yu, J.T., Elishev, A.: On the augmentation topology of automorphism groups of affine spaces and algebras. Int. J. Algebra Comput. 28(8), 1449–1485 (2018). https://doi.org/10.1142/S0218196718400040 Kaliman, S.: Analytic extensions of algebraic isomorphisms. Proc. Amer. Math. Soc. 143(11), 4571–4581 (2015). https://doi.org/10.1090/proc/12684 Kaliman, S., Kutzschebauch, F.: Criteria for the density property of complex manifolds. Invent. Math. 172(1), 71–87 (2008a). https://doi.org/10.1007/s00222-007-0094-6 Kaliman, S., Kutzschebauch, F.: Density property for hypersurfaces \(UV=P(\overline{X})\). Math. Z. 258(1), 115–131 (2008b). https://doi.org/10.1007/s00209-007-0162-z Kaliman, S., Kutzschebauch, F.: On the present state of the Andersén-Lempert theory, Affine algebraic geometry, CRM Proc. Lecture Notes, 54, 85–122, Amer. Math. Soc., Providence, RI (2011) Kaliman, S., Kutzschebauch, F.: On algebraic volume density property. Trans. Groups 21(2), 451–478 (2016). https://doi.org/10.1007/s00031-015-9360-7 Kaliman, S., Kutzschebauch, F.: Algebraic (volume) density property for affine homogeneous spaces. Math. Ann. 367(3–4), 1311–1332 (2017). https://doi.org/10.1007/s00208-016-1451-9 Kaliman, S.: Extensions of isomorphisms of subvarieties in flexible varieties. Transform. Groups 25(2), 517–575 (2020). https://doi.org/10.1007/s00031-019-09546-3 Kaliman, S., Kutzschebauch, F., Leuenberger, M., Complete algebraic vector fields on affine surfaces. Int. J. Math. 31(3), 2050018 (2020). https://doi.org/10.1142/S0129167X20500184 Kutzschebauch, F., Lind, A.: Holomorphic automorphisms of Danielewski surfaces I–density of the group of overshears. Proc. Amer. Math. Soc. 139(11), 3915–3927 (2011). https://doi.org/10.1090/S0002-9939-2011-10855-4 Kutzschebauch, F., Løw, E., Wold, E.F.: Embedding some Riemann surfaces into \(\mathbb{C}^2\) with interpolation. Math. Z., 262(3), 603–611 (2009). https://doi.org/10.1007/s00209-008-0392-8 Kutzschebauch, F.: Flexibility properties in complex analysis and affine algebraic geometry, Automorphisms in birational and affine geometry. Springer Proc. Math. Stat., 79, 387–405, Springer, Cham (2014). https://doi.org/10.1007/978-3-319-05681-4_22 Lárusson, F., Ritter, T.: Proper holomorphic immersions in homotopy classes of maps from finitely connected planar domains into \(\mathbb{C} \times \mathbb{C} ^*\). Indiana Univ. Math. J. 63(2), 367–383 (2014). https://doi.org/10.1512/iumj.2014.63.5206 Leuenberger, M.: (Volume) density property of a family of complex manifolds including the Koras-Russell cubic threefold. Proc. Amer. Math. Soc. 144(9), 3887–3902 (2016). https://doi.org/10.1090/proc/13030 Makar-Limanov, L. On the hypersurface \(x+x^2y+z^2+t^3=0\) in \({\bf C}^4\) or a \({\bf C}^3\)-like threefold which is not \({\bf C}^3\). Israel J. Math. 96(part B), 419–429 (1996). https://doi.org/10.1007/BF02937314 Rosay, J.-P., Rudin, W.: Holomorphic maps from \({ C}^n\) to \({ C}^n\). Trans. Amer. Math. Soc. 310(1), 47–86 (1988). https://doi.org/10.2307/2001110 Ritter, T.: A soft Oka principle for proper holomorphic embeddings of open Riemann surfaces into \((\mathbb{C} ^*)^2\). J. Reine Angew. Math. 745, 59–82 (2018). https://doi.org/10.1515/crelle-2015-0116 Shestakov, I. P., Umirbaev, U. U. The Nagata automorphism is wild. Proc. Natl. Acad. Sci. USA 100(22), 12561–12563 (2003) Snow, D.M.: Stein quotients of connected complex Lie groups. Manus. Math. 50, 185–214 (1985). https://doi.org/10.1007/BF01168831 Springer, T. A. Linear algebraic groups. Prog. Math. 9, 2, Birkhäuser Boston, Inc., Boston, MA (1998). https://doi.org/10.1007/978-0-8176-4840-4 Stolzenberg, G.: Uniform approximation on smooth curves. Acta Math. 115, 185–198 (1966). https://doi.org/10.1007/BF02392207 Stout, E. L. Bounded holomorphic functions on finite Riemann surfaces. Trans. Am. Math. Soc. 120, 255–285 (1965). https://doi.org/10.2307/1994021 Toth, A., Varolin, D.: Holomorphic diffeomorphisms of complex semisimple Lie groups. Invent. Math. 139(2), 351–369 (2000). https://doi.org/10.1007/s002229900029 Ugolini, R., Winkelmann, J.: The Density Property for Vector Bundles (2022). arXiv:2209.05763 Varolin, D.: A general notion of shears, and applications. Michigan Math. J. 46(3), 533–553 (1999). https://doi.org/10.1307/mmj/1030132478 Varolin, D.: The density property for complex manifolds and geometric structures. J. Geom. Anal. 11(1), 135–160 (2001). https://doi.org/10.1007/BF02921959 Vinberg, È. B., Popov, V. L.: Invariant theory, Russian, Algebraic geometry, 4 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow 137–314, 315 (1989) Winkelmann, J.: Invariant rings and quasiaffine quotients. Math. Z. 244(1), 163–174 (2003). https://doi.org/10.1007/s00209-002-0484-9 Wold, E.F.: Proper holomorphic embeddings of finitely and some infinitely connected subsets of \(\mathbb{C} \) into \(\mathbb{C} ^2\). Math. Z. 252(1), 1–9 (2006). https://doi.org/10.1007/s00209-005-0836-3 Zariski, O.: Interprétations algébrico-géométriques du quatorzième problème de Hilbert, French. Bull. Sci. Math. 78(2), 155–168 (1954)