Inventiones mathematicae

, 174:463 | Cite as

Double solid twistor spaces: the case of arbitrary signature

Article

Abstract

In a recent paper ([9]) we constructed a series of new Moishezon twistor spaces which are a kind of variant of the famous LeBrun twistor spaces. In this paper we explicitly give projective models of another series of Moishezon twistor spaces on n CP 2 for arbitrary n≥3, which can be regarded as a generalization of the twistor spaces of ‘double solid type’ on 3CP 2 studied by Kreußler, Kurke, Poon and the author. Similarly to the twistor spaces of ‘double solid type’ on 3CP 2, projective models of the present twistor spaces have a natural structure of double covering of a CP 2-bundle over CP 1. We explicitly give a defining polynomial of the branch divisor of the double covering, whose restriction to fibers is degree four. If n≥4 these are new twistor spaces, to the best of the author’s knowledge. We also compute the dimension of the moduli space of these twistor spaces. Differently from [9], the present investigation is based on analysis of pluri-(half-)anticanonical systems of the twistor spaces.

Keywords

Irreducible Component Twistor Space Exceptional Divisor Rational Curf Arbitrary Signature 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Atiyah, M., Hitchin, N., Singer, I.: Self-duality in four-dimensional Riemannian geometry. Proc. R. Soc. Lond., Ser. A 362, 425–461 (1978)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Campana, F.: On twistor spaces of the class \(\mathcal{C}\). J. Differ. Geom. 33, 541–549 (1991)MATHMathSciNetGoogle Scholar
  3. 3.
    Campana, F., Kreußler, B.: A conic bundle description of Moishezon twistor spaces without effective divisor of degree one. Math. Z. 229, 137–162 (1998)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Hitchin, N.: Kählerian twistor spaces. Proc. Lond. Math. Soc., III. Ser. 43, 133–150 (1981)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Hitchin, N.: Complex manifolds and Einstein’s equations. In: Twistor Geometry and Non-Linear Systems. Lect. Notes Math., vol. 970, pp. 73–99. Springer, Berlin, Heidelberg (1982)CrossRefGoogle Scholar
  6. 6.
    Hitchin, N.: Monopoles and geodesics. Commun. Math. Phys. 83, 579–602 (1982)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Honda, N.: Equivariant deformations of LeBrun’s self-dual metric with torus action. Proc. Am. Math. Soc. 135, 495–505 (2007)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Honda, N.: Self-dual metrics and twenty-eight bitangents. J. Differ. Geom. 75, 175–258 (2007)MATHMathSciNetGoogle Scholar
  9. 9.
    Honda, N.: Explicit construction of new Moishezon twistor spaces. math.DG/0701278 (preprint)Google Scholar
  10. 10.
    Honda, N.: New examples of minitwistor spaces and their moduli space. math.DG/0508088 (preprint)Google Scholar
  11. 11.
    Kreußler, B.: Small resolutions of double solids, branched over a 13-nodal quartic surfaces. Ann. Global Anal. Geom. 7, 227–267 (1989)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Kreußler, B., Kurke, H.: Twistor spaces over the connected sum of 3 projective planes. Compos. Math. 82, 25–55 (1992)MATHGoogle Scholar
  13. 13.
    LeBrun, C.: Explicit self-dual metrics on \(<Emphasis Type="Bold">CP</Emphasis>^2\#\cdots\#<Emphasis Type="Bold">CP</Emphasis>^2\). J. Differ. Geom. 34, 223–253 (1991)MATHMathSciNetGoogle Scholar
  14. 14.
    Pedersen, H., Poon, Y.S.: Self-duality and differentiable structures on the connected sum of complex projective planes. Proc. Am. Math. Soc. 121, 859–864 (1994)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Poon, Y.S.: Compact self-dual manifolds of positive scalar curvature. J. Differ. Geom. 24, 97–132 (1986)MATHMathSciNetGoogle Scholar
  16. 16.
    Poon, Y.S.: On the algebraic structure of twistor spaces. J. Differ. Geom. 36, 451–491 (1992)MATHMathSciNetGoogle Scholar
  17. 17.
    Sommese, A.J.: Some examples of C *-actions. In: Group Actions and Vector Fields. Lect. Notes Math., vol. 956, pp. 118–124. Springer, Berlin, Heidelberg (1982)CrossRefGoogle Scholar
  18. 18.
    Taubes, C.: The existence of anti-self-dual conformal structures. J. Differ. Geom. 36, 163–253 (1992)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsGraduate School of Science and Engineering, Tokyo Institute of TechnologyMeguroJapan

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