Skip to main content
SpringerLink
Log in

On a characterization of analytic compactifications for ℂ* × ℂ*

  • Published:
Bulletin of the Brazilian Mathematical Society, New Series Aims and scope Submit manuscript

Abstract

Let M be a minimal compact surface, let Γ ⊂ M be a compact analytic sub-variety. Assume that X:= M \ Γ is Stein. Then we will show that X admits algebraic compactifications M i (resp. non algebraic compactifications \( \mathbb{M}_i \)) which are not birationally equivalent (resp. not bimeromorphically equivalent) iff X is biholomorphic to

:= ℂ* × ℂ*, a toric surface. However in contrast with

, we shall show that there exist compactifiable Stein surfaces which do not admit any affine structure. Also as applications, we shall characterize the algebraic structures of arbitrary compactifiable surfaces X according to the topological type of Γ.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Atiyah. Complex fibre bundles and Ruled surfaces. Proc. London Math. Soc., 5 (1955), 407–452.

    Article  MATH  MathSciNet  Google Scholar 

  2. A. Beauville. L’application canonique pour les surfaces de type général. Invent. Math., 55 (1979), 121–140.

    Article  MATH  MathSciNet  Google Scholar 

  3. I. Enoki. On surfaces of class V I I 0 with curves. Proc. Jap. Acad., 56 (1980), 275–279.

    Article  MATH  MathSciNet  Google Scholar 

  4. I. Enoki. Surfaces of class V I I 0 with curves. Tohoku Math. J., 33 (1981), 483–492.

    Article  MathSciNet  Google Scholar 

  5. T. Fujita. On the topology of non complete algebraic surfaces. J. Fac. Sci. Uni. Tokyo, 29 (1982), 503–566.

    MATH  Google Scholar 

  6. R.V. Gurjar and M. Miyanishi. On the Jacobian conjecture for Q-homology planes. J. Reine Ange. Math., 516 (1999), 115–132.

    MATH  MathSciNet  Google Scholar 

  7. R. Hartshorne. Ample subvarieties of Algebraic varieties. Lec. Notes in Math. 156 Springer Verlag (1970).

  8. R. Hartshorne. Algebraic Geometry. Graduate Texts in Mathematics, 52, Springer Verlag (1977).

  9. A. Howard. On the compactification of a Stein surface. Math. Ann., 176 (1968), 221–224.

    Article  MATH  MathSciNet  Google Scholar 

  10. S. Iitaka. On logarithmic K-3 surfaces. Osaka J. Math., 16 (1979), 675–705.

    MATH  MathSciNet  Google Scholar 

  11. S. Iitaka. On logarithmic Kodaira dimension of algebraic varieties, in Complex Analysis and Algebraic Geometry; Tokyo, Iwanami, (1977), 115–189.

  12. S. Iitaka. Algebraic geometry. Graduate Text in Math., Springer Verlag, 79 (1981).

  13. Y. Imayoshi. Holomorphic family of Riemann surfaces and Teichmuller spaces II. Tohoku Math. J., 31 (1979), 469–489.

    Article  MATH  MathSciNet  Google Scholar 

  14. M. Inoue. New surfaces with no meromorphic functions. Proc. Int. Congress of Math. Vancouver 1974. Vol. 1 (R. James, ed.), (1975), 423–426.

  15. Ma. Kato. Compact complex surfaces manifolds containing “global” spherical shells. Inter. Sympo. in Algebraic geometry, Kypto (1977), 45–84.

  16. K. Kodaira. On the structure of complex analytic surfaces II. Amer. J. Math., 90 (1966), 682–721.

    Article  MathSciNet  Google Scholar 

  17. K. Kodaira. Holomorphic mappings of polydiscs into compact complex manifolds. J. Differ. Geometry, 6 (1971/72), 33–46.

    MATH  MathSciNet  Google Scholar 

  18. Y. Matsushima and A. Morimoto. Sur certaines espaces fbrés holomorphes sur une variété de Stein. Bull. Soc. Math. France, 88 (1960), 137–155.

    MATH  MathSciNet  Google Scholar 

  19. N. Mohan Kumar. Affne like surfaces. J. Alg. Geom., 2 (1993), 689–703.

    MATH  MathSciNet  Google Scholar 

  20. A. Neeman. Ueda Theory: Theorems and Problems. Memoirs AMS, 415 (1989).

  21. F. Sakai. Kodaira dimensions of complements of divisors, in Complex Analysis and Algebraic Geometry, Tokyo, Iwanami (1977), 239–257.

  22. G.K. Sankaran. Remarks on compact surfaces. Osaka J. Math., 29 (1992), 63–70.

    MATH  MathSciNet  Google Scholar 

  23. J.P. Serre. Groupes algébriques et corps de classes. Hermann, Paris (1959).

    MATH  Google Scholar 

  24. A. Shastri. Compact structures on C* × C*. Tohoku Math. J., 40 (1988), 35–49.

    Article  MATH  MathSciNet  Google Scholar 

  25. R. Simha. Algebraic varieties bihomorphic to C* × C*. Tohoku Math. J., 30 (1978), 455–461.

    Article  MATH  MathSciNet  Google Scholar 

  26. T. Suwa. On ruled surfaces of genus 1. J. Math. Soc. Japan, 21 (1969), 291–311.

    Article  MATH  MathSciNet  Google Scholar 

  27. T. Ueda. Compactifications of C × C* and C*2. Tohoku Math. J., 31 (1979), 81–90.

    Article  MATH  MathSciNet  Google Scholar 

  28. T. Vo Van. On the compactification problem for strongly pseudoconvex surfaces. Proc. AMS, 82 (1981), 407–410.

    MATH  Google Scholar 

  29. T. Vo Van. On the compactification problems for strongly pseudoconvex surfaces III. Math. Zeit., 195 (1987), 259–267.

    Article  MATH  Google Scholar 

  30. T. Vo Van. On the compactification problems for Stein surfaces. Compo. Mathe-matica, 71 (1989), 1–12.

    MATH  Google Scholar 

  31. T. Vo Van. On the compactification problems for Stein threefolds. Proc. Sympo. in Pure Math., 52, Part II (1991), 535–542.

    Google Scholar 

  32. T. Vo Van. On the problems of Hartshorne and Serre for some C-analytic surfaces. C.R. Acad. Sci., 326 (1998), 465–470.

    MATH  Google Scholar 

  33. T. Vo Van. On Hartshorne’s problem for compact C-analytic surfaces with k(M) = −∞. Bull. Sci. Math., 123 (1999), 623–641.

    Article  MathSciNet  Google Scholar 

  34. T. Vo Van. An analogue of Hatrshorne and Serre problems for 1-convex surfaces. Bull. Sci. Math., 127 (2003), 37–54.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Suffolk University, Beacon Hill, Boston, Mass., 02114, USA

    Vo Van Tan

Authors
  1. Vo Van Tan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vo Van Tan.

About this article

Cite this article

Van Tan, V. On a characterization of analytic compactifications for ℂ* × ℂ*. Bull Braz Math Soc, New Series 41, 355–387 (2010). https://doi.org/10.1007/s00574-010-0016-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00574-010-0016-x

Keywords

Mathematical subject classification

Search

Navigation