Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 


(Volume) density property of a family of complex manifolds including the Koras-Russell cubic threefold

Author: Matthias Leuenberger
Journal: Proc. Amer. Math. Soc. 144 (2016), 3887-3902
MSC (2010): Primary 32M05, 32M25, 14R10
Published electronically: March 17, 2016
MathSciNet review: 3513546
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We present modified versions of existing criteria for the density property and the volume density property of complex manifolds. We apply these methods to show the (volume) density property for a family of manifolds given by $ x^2y=a(\bar z) + xb(\bar z)$ with $ \bar z =(z_0,\ldots ,z_n)\in \mathbb{C}^{n+1}$ and holomorphic volume form $ \mathrm {d} x/x^2\wedge \mathrm {d} z_0\wedge \ldots \wedge \mathrm {d} z_n$. The key step is to show that in certain cases transitivity of the action of (volume preserving) holomorphic automorphisms implies the (volume) density property, and then to give sufficient conditions for the transitivity of this action. In particular, we show that the Koras-Russell cubic threefold $ \lbrace x^2y + x + z_0^2 + z_1^3=0\rbrace $ has the density property and the volume density property.

References [Enhancements On Off] (What's this?)

  • [1] Erik Andersén and László Lempert, On the group of holomorphic automorphisms of $ {\bf C}^n$, Invent. Math. 110 (1992), no. 2, 371-388. MR 1185588 (93i:32038),
  • [2] A. Dubouloz, L. Moser-Jauslin, and P.-M. Poloni, Inequivalent embeddings of the Koras-Russell cubic 3-fold, Michigan Math. J. 59 (2010), no. 3, 679-694. MR 2745757 (2011m:14103),
  • [3] Franc Forstnerič, Stein manifolds and holomorphic mappings: The homotopy principle in complex analysis, 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], vol. 56, Springer, Heidelberg, 2011. MR 2975791
  • [4] Franc Forstnerič and Jean-Pierre Rosay, Approximation of biholomorphic mappings by automorphisms of $ {\bf C}^n$, Invent. Math. 112 (1993), no. 2, 323-349. MR 1213106 (94f:32032),
  • [5] Lars Hörmander, An introduction to complex analysis in several variables, 3rd ed., North-Holland Mathematical Library, vol. 7, North-Holland Publishing Co., Amsterdam, 1990. MR 1045639 (91a:32001)
  • [6] Shulim Kaliman and Frank Kutzschebauch, Criteria for the density property of complex manifolds, Invent. Math. 172 (2008), no. 1, 71-87. MR 2385667 (2008k:32066),
  • [7] Shulim Kaliman and Frank Kutzschebauch, Algebraic volume density property of affine algebraic manifolds, Invent. Math. 181 (2010), no. 3, 605-647. MR 2660454 (2011k:32031),
  • [8] Shulim Kaliman and Frank Kutzschebauch, On the present state of the Andersén-Lempert theory, Affine algebraic geometry, CRM Proc. Lecture Notes, vol. 54, Amer. Math. Soc., Providence, RI, 2011, pp. 85-122. MR 2768636
  • [9] S. Kaliman and F. Kutzschebauch,
    On algebraic volume density property,
    ArXiv e-prints, Jan. 2012.
    arXiv1201.4769, to appear in Transfom. Groups.
  • [10] Sh. Kaliman and M. Zaidenberg, Affine modifications and affine hypersurfaces with a very transitive automorphism group, Transform. Groups 4 (1999), no. 1, 53-95. MR 1669174 (2000f:14099),
  • [11] Frank Kutzschebauch, Flexibility properties in complex analysis and affine algebraic geometry, Automorphisms in birational and affine geometry, Springer Proc. Math. Stat., vol. 79, Springer, Cham, 2014, pp. 387-405. MR 3229363,
  • [12] F. Kutzschebauch and S. Kaliman,
    On Density and Volume Density Property,
    Complex Analysis and Geometry, 2014,
    Springer Proceedings in Mathematics & Statistics, KSCV10, Gyeongju, Korea.
  • [13] F. Kutzschebauch and M. Leuenberger,
    Lie algebra generated by locally nilpotent derivations on Danielewski surfaces,
    ArXiv e-prints, Nov. 2013,
    arXiv1311.1075, to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci. (5).
  • [14] L. Makar-Limanov, 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$. part B, Israel J. Math. 96 (1996), no. part B, 419-429. MR 1433698 (98a:14052),
  • [15] P.-M. Poloni, Classification(s) of Danielewski hypersurfaces, Transform. Groups 16 (2011), no. 2, 579-597. MR 2806502 (2012e:14115),
  • [16] Árpád Tóth and Dror Varolin, Holomorphic diffeomorphisms of semisimple homogeneous spaces, Compos. Math. 142 (2006), no. 5, 1308-1326. MR 2264667 (2007i:32022),
  • [17] Dror Varolin, The density property for complex manifolds and geometric structures, J. Geom. Anal. 11 (2001), no. 1, 135-160. MR 1829353 (2002g:32026),

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 32M05, 32M25, 14R10

Retrieve articles in all journals with MSC (2010): 32M05, 32M25, 14R10

Additional Information

Matthias Leuenberger
Affiliation: Mathematisches Institut, Universität Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland

Keywords: Density property, holomorphic automorphisms, Koras-Russell cubic threefold
Received by editor(s): July 14, 2015
Received by editor(s) in revised form: November 1, 2015, and November 5, 2015
Published electronically: March 17, 2016
Additional Notes: The author was partially supported by Schweizerischer Nationalfond Grant 200021-140235/1
Communicated by: Franc Forstneric
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society