Holomorphic Flexibility Properties of the Space of Cubic Rational Maps

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• 作者：Alexander Hanysz
• 关键词：Stein manifold ; Oka manifold ; Rational function ; Holomorphic flexibility ; Cross ; ratio ; Geometric invariant theory ; Categorical quotient ; $${\mathbb C}$$ C ; connected ; Dominable ; Strongly dominable ; Primary 32Q28 ; Secondary 32H02 ; 32Q55 ; 54C35 ; 58D15
• 刊名：Journal of Geometric Analysis
• 出版年：2015
• 出版时间：July 2015
• 年：2015
• 卷：25
• 期：3
• 页码：1620-1649
• 全文大小：580 KB
• 参考文献：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/-303.-804
  2.Buzzard, G.T., Lu, S.S.Y.: Algebraic surfaces holomorphically dominable by \({\bf C}^2\) . Invent. Math. 139(3), 617-59 (2000)
  3.Campana, F., Winkelmann, J.: On \(h\) -principle and specialness for complex projective manifolds. Preprint at http://?arxiv.?org/?abs/-210.-369
  4.Donaldson, S.: Riemann Surfaces, Oxford Graduate Texts in Mathematics, vol. 22. Oxford University Press, Oxford (2011)
  5.Douady, A.: Le problème des modules pour les sous-espaces analytiques compacts d’un espace analytique donné. Ann. Inst. Fourier (Grenoble) 16(1), 1-5 (1966)
  6.Forstneri?, F.: Oka manifolds. C. R. Math. Acad. Sci. Paris 347, 1017-020 (2009)MATH MathSciNet View Article
  7.Forstneri?, F.: Stein manifolds and holomorphic mappings, Ergeb. Math. Grenzgeb. (3), vol. 56. Springer-Verlag, (2011)
  8.Forstneri?, F., Lárusson, F.: Survey of Oka theory. New York J. Math. 17A, 11-8 (2011)MATH
  9.Gorbatsevich, V.V., Onishchik, A.L., Vinberg, E.B.: Foundations of Lie Theory and Lie Transformation Groups. Springer, Berlin (1997)MATH
  10.Gromov, M.: Oka’s principle for holomorphic sections of elliptic bundles. J. Am. Math. Soc. 2(4), 851-97 (1989)MATH MathSciNet
  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), 631-38 (1995)MATH MathSciNet
  12.Hanysz, A.: Oka properties of some hypersurface complements. Proc. Am. Math. Soc. 142(2), 483-96 (2014)
  13.Havlicek, J.W.: The cohomology of holomorphic self-maps of the Riemann sphere. Math. Z. 218(2), 179-90 (1995)MATH MathSciNet View Article
  14.Jones, G.A., Singerman, D.: Complex Functions: An Algebraic and Geometric Viewpoint. Cambridge University Press, Cambridge (1987)MATH View Article
  15.Kaup, W.: Holomorphic mappings of complex spaces. In: Symposia Mathematica, Vol. II (INDAM, Rome, 1968), pp. 333-40. Academic Press, London (1969)
  16.Lando, S.K., Zvonkin, A.K.: Graphs on Surfaces and their Applications, Encyclopaedia of Mathematical Sciences, vol. 141. Springer-Verlag, Berlin (2004)
  17.Levy, A.: The space of morphisms on projective space. Acta Arith. 146(1), 13-1 (2011)MATH MathSciNet View Article
  18.Luna, D.: Slices étales. In Sur les groupes algébriques, 81-05. Bull. Soc. Math. France, Paris, Mémoire 33. Soc. Math. France, Paris (1973)
  19.Lyndon, R.C., Ullman, J.L.: Groups of elliptic linear fractional transformations. Proc. Am. Math. Soc. 18(6), 1119-124 (1967)MATH MathSciNet View Article
  20.Milnor, J.: Geometry and dynamics of quadratic rational maps. Exp. Math. 2(1), 37-3. With an appendix by the author and Lei Tan (1993)
  21.Ono, Y., Yamaguchi, K.: Group actions on spaces of rational functions. Publ. Res. Inst. Math. Sci. 39(1), 173-81 (2003)MATH MathSciNet View Article
  22.Rainer, A.: Orbit projections as fibrations. Czechoslovak Math. J. 59(2), 529-38 (2009)MATH MathSciNet View Article
  23.Segal, G.: The topology of spaces of rational functions. Acta Math. 143(1), 39-2 (1979)MATH MathSciNet View Article
  24.Silverman, J.H.: The space of rational maps on \({\bf P}^1\) . Duke Math. J. 94(1), 41-7 (1998)
  25.Snow, D.M.: Reductive group actions on Stein spaces. Math. Ann. 259(1), 79-7 (1982)MATH MathSciNet View Article
  
• 作者单位：Alexander Hanysz (1)
  
  1. School of Mathematical Sciences, University of Adelaide, Adelaide, SA, 5005, Australia
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Differential Geometry
  Convex and Discrete Geometry
  Fourier Analysis
  Abstract Harmonic Analysis
  Dynamical Systems and Ergodic Theory
  Global Analysis and Analysis on Manifolds
  
• 出版者：Springer New York
• ISSN：1559-002X

文摘

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 M?bius 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.