Holomorphic flexibility properties of the space of cubic rational maps

TL;DR

This paper investigates the complex geometric structures of the space of cubic rational maps, demonstrating its holomorphic flexibility and describing its quotient structure using geometric invariant theory.

Contribution

It applies geometric invariant theory to analyze the complex structure of degree 2 and 3 rational maps, revealing their Oka properties and explicit quotient descriptions.

Findings

R_2 is an Oka manifold due to transitive M"obius action.

R_3 has a categorical quotient with an explicit formula.

R_3 exhibits strong holomorphic flexibility properties.

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 M\"obius 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 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 C-connectedness.