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Mathematische Annalen

, Volume 297, Issue 1, pp 627–662 | Cite as

The complete classification of compactifications of ℂ3 which are projective manifolds with the second Betti number one

  • Mikio Furushima
Article

Mathematics Subject Classification (1991)

32J05 

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References

  1. [B-K]
    Barthel, G., Kaup, L.: Topologie des espaces complexes compactes singulieres. (Montreal Lect. Notes, vol. 80) Montreal: 1982Google Scholar
  2. [B1]
    Brenton, L.: Some algebraicity criteria for singular surfaces. Invent. Math41, 129–147 (1977)Google Scholar
  3. [B2]
    Brenton, L.: On singular complex surface with negative canonical bundle, with applications to singular compactification of ℂ3 and 3-dimensional rational singularities. Math. Ann.248, 117–124 (1980)Google Scholar
  4. [B-M]
    Brenton, L., Morrow, J.A.: Compactification of ℂ3. Trans. Am. Math. Soc.246, 139–158 (1979)Google Scholar
  5. [C-T]
    Coray, D.F., Tsfasman, M.A.: Arithmetic on singular del Pezzo surfaces. Proc. Lond. Math. Soc.57, 25–87 (1988)Google Scholar
  6. [F]
    Fujita, T.: Classification theories of polarized varieties. (Lond. Math. Soc. Lect. Note Ser., vol. 155) Cambridge: Cambridge University Press 1990Google Scholar
  7. [Fu1]
    Furushima, M.: Singular del Pezzo surfaces and analytic compactifications 3-dimensional complex affine space ℂ3. Nagoya Math. J.104, 1–28 (1986)Google Scholar
  8. [Fu2]
    Furushima, M.: Complex analytic compactifications of ℂ3. Compos. Math.76, 163–196 (1990)Google Scholar
  9. [Fu3]
    Furushima, M.: Mukai-Umemura's example of a Fano threefold of genus 12 as a compactification of ℂ3. Nagoya Math. J.127, 145–165 (1992)Google Scholar
  10. [Fu4]
    Furushima, M.: The structure of compactification of ℂ3. Proc. Japan Acad.68A-(2), 33–36 (1992)Google Scholar
  11. [Fu5]
    Furushima, M.: A new example of a compactification of ℂ3. Math. Z.212, 395–399 (1993)Google Scholar
  12. [F-N1]
    Furushima, M., Nakayama, N.: A new construction of a compactification of ℂ3. Tohoku Math. J.41, 543–560 (1989)Google Scholar
  13. [F-N2]
    Furushima, M., Nakayama, N.: The family of lines on the Fano threefoldV 5. Nagoya Math. J.116, 111–122 (1989)Google Scholar
  14. [F-T]
    Furushima, M., Tada, M.: Non-normal del Pezzo surfaces and Fano threefolds of first kind. J. Reine Angew. Math.429, 183–190 (1992)Google Scholar
  15. [F-O]
    Furushima, M., Oguiso, K.: Two remarks on Moishezon Calabi-Yau-3-folds. MPI 93-15 (Preprint 1993)Google Scholar
  16. [H-W]
    Hidaka, F., Watanabe, K.: Normal Gorenstein surfaces with ample anti-canonical divisor. Tokyo J. Math.4, 319–330 (1981)Google Scholar
  17. [Hi]
    Hirzebruch, F.: Some problems on differentiable and complex manifolds. Ann. Math.60, 213–236 (1954)Google Scholar
  18. [Is1]
    Iskovskih, V.A.: Anticanonical models of three-dimensional algebraic varieties. J. Sov. Math.13–14, 748–814 (1980)Google Scholar
  19. [Is2]
    Iskovskih, V.A.: Double projection from a line on Fano threefold of the first kind. Math. USSR Sb.66, 265–284 (1990)Google Scholar
  20. [K-M-M]
    Kawamata, Y., Matsuda, K., Matsuki, K.: Introduction to minimal model problem. In: Oda, T. (ed.) Algebraic geometry, Sendai. (Adv. Stud. Pure Math., vol. 10, pp. 283–360) Amsterdam: North-Holland and Tokyo: Kinokuniya 1987Google Scholar
  21. [K]
    Kollár, J.: Flops. Nagoya Math. J.113, 15–36 (1989)Google Scholar
  22. [Mo]
    Mori, S.: Threefolds whose canonical bundles are not numerically effective. Ann. Math.116, 133–176 (1982)Google Scholar
  23. [M-U]
    Mukai, S., Umemura, H.: Minimal rational threefolds. In: Raynaud, M., Shioda, I. (eds.) Algebraic geometry. (Lect. Notes Math., vol. 1016, pp. 490–518) Berlin Heidelberg New York: Springer 1983Google Scholar
  24. [M]
    Mukai, S.: On Fano threefolds. In: Projective geometry. Trieste, 1989.Google Scholar
  25. [P-S1]
    Peternell, T., Schneider, M.: Compactifications of ℂn. I. Math. Ann.280, 129–146 (1988)Google Scholar
  26. [P-S2]
    Peternell, T. Schneider, M.: Compactifications of ℂ3n: A survey. In: Bedford, E. et al. (eds.) Several complex variables and complex geometry. (Proc. Symp. Pure Math., vol. 52, part 2. pp. 455–466) Providence, RI: Am. Math. Soc. 1991Google Scholar
  27. [P]
    Peternell, T.: Compactifications of ℂ3. II. Math. Ann.283, 121–137 (1989)Google Scholar
  28. [Pr]
    Prokhorov, Yu.G.: Fano threefolds of genus 12 and compactifications of ℂ3. Leningr. Math. J.3, 162–170 (1991)Google Scholar
  29. [Re]
    Reid, M.: Minimal model of canonical 3-folds. In: Iitaka, S. (ed.) Algebraic varieties and analytic varieties. (Adv. Stud. Pure Math., vol. 1, pp. 131–180) Amsterdam: North-Holland and Tokyo: Kinokuniya 1981Google Scholar
  30. [R]
    Reider, I.: Vector bundles of rank 2 and linear systems on algebraic surfaces. Ann. Math.127, 309–316 (1988)Google Scholar
  31. [S]
    Sakai, F.: Reider-Serrano's method on normal surfaces. In: Sommese, A.J. et al. (eds.) Algebraic geometry. (Lect. Notes Math., vol. 1417, pp. 301–309) Berlin Heidelberg New York: Springer 1990Google Scholar
  32. [Sh]
    Shokulov, V.V.: Smoothness of the general anticanonical divisor on a Fano 3-fold. Math. USSR, Izv.14, 395–405 (1980)Google Scholar
  33. [T]
    Takeuchi, K.: Some birational maps of Fano 3-folds. Compos. Math.71, 265–283 (1989)Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Mikio Furushima
    • 1
  1. 1.Department of Mathematics, College of EducationRyukyu UniversityOkinawaJapan

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