Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Complex Manifolds

Ed. by Fino, Anna Maria

Covered by Web of Science - Emerging Sources Citation Index and Zentralblatt Math (zbMATH)

CiteScore 2018: 0.64

SCImago Journal Rank (SJR) 2018: 0.643
Source Normalized Impact per Paper (SNIP) 2018: 0.812

Mathematical Citation Quotient (MCQ) 2018: 0.61

Open Access
See all formats and pricing
More options …

Geometry of some twistor spaces of algebraic dimension one

Nobuhiro Honda
Published Online: 2015-09-09 | DOI: https://doi.org/10.1515/coma-2015-0009


It is shown that there exists a twistor space on the n-fold connected sum of complex projective planes nCP2, whose algebraic dimension is one and whose general fiber of the algebraic reduction is birational to an elliptic ruled surface or a K3 surface. The former kind of twistor spaces are constructed over nCP2 for any n ≥ 5, while the latter kind of example is constructed over 5CP2. Both of these seem to be the first such example on nCP2. The algebraic reduction in these examples is induced by the anti-canonical system of the twistor spaces. It is also shown that the former kind of twistor spaces contain a pair of non-normal Hopf surfaces.

MSC: 53A30


  • Google Scholar

  • [1] M. Atiyah, N. Hitchin, I. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London, Ser. A 362 (1978) 425–461. Google Scholar

  • [2] F. Campana, The class C is not stable by small deformations, Math. Ann. 229 (1991) 19–30. Google Scholar

  • [3] I. Enoki, Surfaces of class VII0 with curves, Tohoku Math. J. 33 (1981) 453–492. Google Scholar

  • [4] S. K. Donaldson, R. Friedman, Connected sums of self-dual manifolds and deformations of singular spaces Non-linearlity 2 (1989) 197–239. Google Scholar

  • [5] R. Friedman, D. Morrison, The birational geometry of degenerations: An overview, The birational geometry of degenerations (R. Friedman, D. Morrison, eds.) Progress Math. 29 (1983) 1–32. Google Scholar

  • [6] A. Fujiki. On the structure of compact complex manifolds in C, Adv. Stud. Pure Math. 1 (1983) 231–302. Google Scholar

  • [7] A. Fujiki. Compact self-dual manifolds with torus actions, J. Differential Geom. 55 (2000) 229–324. Google Scholar

  • [8] A. Fujiki, Algebraic reduction of twistor spaces of Hopf surfaces, Osaka J. Math. 37 (2000) 847–858. Google Scholar

  • [9] P. Griffiths and J. Harris, “Principle of Algebraic Geometry”, Wiley-Interscience. Google Scholar

  • [10] J. Hausen, Zur Klassifikation of glatter kompakter C*-Flachen, Math. Ann. 301 (1995), 763–769. Google Scholar

  • [11] N. Hitchin, Linear field equations on self-dual spaces, Proc. Roy. Soc. London Ser. A 370 (1980) 173-191. Google Scholar

  • [12] N. Hitchin, K¨ahlerian twistor spaces, Proc. London Math. Soc. (3) 43 (1981) 133-150. CrossrefGoogle Scholar

  • [13] N. Honda and M. Itoh, A Kummer type construction of self-dual metrics on the connected sum of four complex projective planes, J. Math. Soc. Japan 52 (2000) 139-160. CrossrefGoogle Scholar

  • [14] N. Honda, Double solid twistor spaces II: general case, J. reine angew. Math. 698 (2015) 181-220. Google Scholar

  • [15] E. Horikawa, Deformations of holomorphic maps III, Math. Ann. 222 (1976) 275–282. Google Scholar

  • [16] M. Inoue, New surfaces with no meromorphic functions, Proc. Int. Cong. Math., Vancouver 1 (1974), 423–426. Google Scholar

  • [17] D. Joyce, Explicit construction of self-dual 4-manifolds, Duke Math. J. 77 (1995) 519-552. Google Scholar

  • [18] B. Kreußler and H. Kurke, Twistor spaces over the connected sum of 3 projective planes. Compositio Math. 82:25–55, 1992. Google Scholar

  • [19] C. LeBrun, Y. Poon Twistors, K¨ahler manifolds, and bimeromorphic geometry II, J. Amer. Math. Soc. 5 (1992), 317–325. Google Scholar

  • [20] K. Nishiguchi Degenerations of K3 surfaces, J. Math. Kyoto Univ. 82 (1988) 267–300. Google Scholar

  • [21] P. Orlik, P. Wagreich, Algebraic surfaces with k*-actions, Acta Math. 138 (1977) 43–81. Google Scholar

  • [22] H. Pedersen, Y. S. Poon, Self-duality and differentiable structures on the connected sum of complex projective planes, Proc. Amer. Math. Soc. 121 (1994) 859-864. Google Scholar

  • [23] Y. S. Poon, On the algebraic structure of twistor spaces, J. Differential Geom. 36 (1992), 451–491. Google Scholar

  • [24] K. Ueno, Classification theory of algebraic varieties and compact complex spaces, Lect. Note Math. 439 (1975) Google Scholar

  • [25] O. Zariski, The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surfaces, Ann. Math. 76 (1962), 560–615. Google Scholar

About the article

Received: 2015-06-02

Accepted: 2015-08-29

Published Online: 2015-09-09

Citation Information: Complex Manifolds, Volume 2, Issue 1, ISSN (Online) 2300-7443, DOI: https://doi.org/10.1515/coma-2015-0009.

Export Citation

© 2015 Nobuhiro Honda . This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Comments (0)

Please log in or register to comment.
Log in