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The Journal of Geometric Analysis

, Volume 25, Issue 3, pp 1620–1649 | Cite as

Holomorphic Flexibility Properties of the Space of Cubic Rational Maps

  • Alexander Hanysz
Article

Abstract

For each natural number \(d\), the space \(R_d\) of rational maps of degree \(d\) on the Riemann sphere has the structure of a complex manifold. The topology of these manifolds has been extensively studied. The recent development of Oka theory raises some new and interesting questions about their complex structure. We apply geometric invariant theory to the cases of degree 2 and 3, studying a double action of the Möbius group on \(R_d\). The action on \(R_2\) is transitive, implying that \(R_2\) is an Oka manifold. The action on \(R_3\) has \({\mathbb C}\) as a categorical quotient; we give an explicit formula for the quotient map and describe its structure in some detail. We also show that \(R_3\) enjoys the holomorphic flexibility properties of strong dominability and \({\mathbb C}\)-connectedness.

Keywords

Stein manifold Oka manifold Rational function   Holomorphic flexibility Cross-ratio Geometric invariant theory  Categorical quotient \({\mathbb C}\)-connected Dominable Strongly dominable 

Mathematics Subject Classification

Primary 32Q28 Secondary 32H02 32Q55 54C35 58D15 

Notes

Acknowledgments

I thank Finnur Lárusson for many helpful discussions during the preparation of this paper.

References

  1. 1.
    Andrist, R.B., Wold, E.F.: The complement of the closed unit ball in \(\mathbb{C}^3\) is not subelliptic. Preprint at http://arxiv.org/abs/1303.1804
  2. 2.
    Buzzard, G.T., Lu, S.S.Y.: Algebraic surfaces holomorphically dominable by \({\bf C}^2\). Invent. Math. 139(3), 617–659 (2000)Google Scholar
  3. 3.
    Campana, F., Winkelmann, J.: On \(h\)-principle and specialness for complex projective manifolds. Preprint at http://arxiv.org/abs/1210.7369
  4. 4.
    Donaldson, S.: Riemann Surfaces, Oxford Graduate Texts in Mathematics, vol. 22. Oxford University Press, Oxford (2011)Google Scholar
  5. 5.
    Douady, A.: Le problème des modules pour les sous-espaces analytiques compacts d’un espace analytique donné. Ann. Inst. Fourier (Grenoble) 16(1), 1–95 (1966)Google Scholar
  6. 6.
    Forstnerič, F.: Oka manifolds. C. R. Math. Acad. Sci. Paris 347, 1017–1020 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Forstnerič, F.: Stein manifolds and holomorphic mappings, Ergeb. Math. Grenzgeb. (3), vol. 56. Springer-Verlag, (2011)Google Scholar
  8. 8.
    Forstnerič, F., Lárusson, F.: Survey of Oka theory. New York J. Math. 17A, 11–38 (2011)zbMATHGoogle Scholar
  9. 9.
    Gorbatsevich, V.V., Onishchik, A.L., Vinberg, E.B.: Foundations of Lie Theory and Lie Transformation Groups. Springer, Berlin (1997)zbMATHGoogle Scholar
  10. 10.
    Gromov, M.: Oka’s principle for holomorphic sections of elliptic bundles. J. Am. Math. Soc. 2(4), 851–897 (1989)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Guest, M.A., Kozlowski, A., Murayama, M., Yamaguchi, K.: The homotopy type of the space of rational functions. J. Math. Kyoto Univ. 35(4), 631–638 (1995)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Hanysz, A.: Oka properties of some hypersurface complements. Proc. Am. Math. Soc. 142(2), 483–496 (2014)Google Scholar
  13. 13.
    Havlicek, J.W.: The cohomology of holomorphic self-maps of the Riemann sphere. Math. Z. 218(2), 179–190 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Jones, G.A., Singerman, D.: Complex Functions: An Algebraic and Geometric Viewpoint. Cambridge University Press, Cambridge (1987)zbMATHCrossRefGoogle Scholar
  15. 15.
    Kaup, W.: Holomorphic mappings of complex spaces. In: Symposia Mathematica, Vol. II (INDAM, Rome, 1968), pp. 333–340. Academic Press, London (1969)Google Scholar
  16. 16.
    Lando, S.K., Zvonkin, A.K.: Graphs on Surfaces and their Applications, Encyclopaedia of Mathematical Sciences, vol. 141. Springer-Verlag, Berlin (2004)Google Scholar
  17. 17.
    Levy, A.: The space of morphisms on projective space. Acta Arith. 146(1), 13–31 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Luna, D.: Slices étales. In Sur les groupes algébriques, 81–105. Bull. Soc. Math. France, Paris, Mémoire 33. Soc. Math. France, Paris (1973)Google Scholar
  19. 19.
    Lyndon, R.C., Ullman, J.L.: Groups of elliptic linear fractional transformations. Proc. Am. Math. Soc. 18(6), 1119–1124 (1967)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Milnor, J.: Geometry and dynamics of quadratic rational maps. Exp. Math. 2(1), 37–83. With an appendix by the author and Lei Tan (1993)Google Scholar
  21. 21.
    Ono, Y., Yamaguchi, K.: Group actions on spaces of rational functions. Publ. Res. Inst. Math. Sci. 39(1), 173–181 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Rainer, A.: Orbit projections as fibrations. Czechoslovak Math. J. 59(2), 529–538 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Segal, G.: The topology of spaces of rational functions. Acta Math. 143(1), 39–72 (1979)zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Silverman, J.H.: The space of rational maps on \({\bf P}^1\). Duke Math. J. 94(1), 41–77 (1998)Google Scholar
  25. 25.
    Snow, D.M.: Reductive group actions on Stein spaces. Math. Ann. 259(1), 79–97 (1982)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2014

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of AdelaideAdelaideAustralia

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