Abstract
We introduce the notion of algebraic volume density property for affine algebraic manifolds and prove some important basic facts about it, in particular that it implies the volume density property. The main results of the paper are producing two big classes of examples of Stein manifolds with volume density property. One class consists of certain affine modifications of ℂn equipped with a canonical volume form, the other is the class of all Linear Algebraic Groups equipped with the left invariant volume form.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andersén, E.: Volume-preserving automorphisms of ℂn. Complex Var. Theory Appl. 14(1–4), 223–235 (1990)
Andersén, E.: Complete vector fields on (ℂ) n. Proc. Am. Math. Soc. 128(4), 1079–1085 (2000)
Andersén, E., Lempert, L.: On the group of holomorphic automorphisms of ℂn. Invent. Math. 110(2), 371–388 (1992)
Bourbaki, N.: Elements of Mathematics. Lie Groups and Lie Algebras. Chaps. 7–9. Springer, Berlin (2005)
Campana, F.: On twistor spaces of the class C. J. Differ. Geom. 33, 541–549 (1991)
Donzelli, F., Dvorsky, A., Kaliman, S.: Algebraic density property of homogeneous spaces. Trans. Groups (to appear)
Forstnerič, F.: A theorem in complex symplectic geometry. J. Geom. Anal. 5(3), 379–393 (1995)
Forstnerič, F., Rosay, J.-P.: Approximation of biholomorphic mappings by automorphisms of ℂn. Invent. Math. 112(2), 323–349 (1993)
Grothendieck, A.: On the de Rham cohomology of algebraic varieties. Inst. Hautes Etud. Sci. Publ. Math. 29, 95–105 (1966)
Hartshorne, R.: Algebraic Geometry. Springer, New York (1977). 496 pp.
Hadžiev, Dž.: Certain questions of the theory of vector invariants. Mat. Sb. 72, 420–435 (1967) (Russian)
Kaliman, S., Kutzschebauch, F.: Criteria for the density property of complex manifolds. Invent. Math. 172(1), 71–87 (2008)
Kaliman, S., Kutzschebauch, F.: Density property for hypersurfaces \(uv=p({\bar{x}})\). Math. Z. 258(1), 115–131 (2008)
Kaliman, S., Zaidenberg, M.: Affine modifications and affine hypersurfaces with a very transitive automorphism group. Transform. Groups 4, 53–95 (1999)
Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. I. Wiley, New York (1996). Reprint of the 1963 original. Wiley Classics Library. A Wiley-Interscience Publication
Kollár, J., Miyaoka, Y., Mori, S.: Rationally connected varieties. J. Algebraic Geom. 1, 429–448 (1992)
Onishchik, A.L., Vinberg, E.B.: Lie Groups and Algebraic Groups. Springer Series in Soviet Mathematics. Springer, Berlin (1996)
Mostow, G.D.: Fully reducible subgroups of algebraic groups. Am. J. Math. 78, 200–221 (1956)
Serre, J.P.: Espaces fibres algébriques. Séminaire C. Chevalley 3(1), 1–37 (1958)
Toth, A., Varolin, D.: Holomorphic diffeomorphisms of complex semisimple Lie groups. Invent. Math. 139(2), 351–369 (2000)
Toth, A., Varolin, D.: Holomorphic diffeomorphisms of semisimple homogenous spaces. Compos. Math. 142(5), 1308–1326 (2006)
Varolin, D.: The density property for complex manifolds and geometric structures. J. Geom. Anal. 11(1), 135–160 (2001)
Varolin, D.: The density property for complex manifolds and geometric structures II. Int. J. Math. 11(6), 837–847 (2000)
Varolin, D.: A general notion of shears, and applications. Mich. Math. J. 46(3), 533–553 (1999)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kaliman, S., Kutzschebauch, F. Algebraic volume density property of affine algebraic manifolds. Invent. math. 181, 605–647 (2010). https://doi.org/10.1007/s00222-010-0255-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00222-010-0255-x