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 HanyszEmail author


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.


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 



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


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Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of AdelaideAdelaideAustralia

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