On Real and Pseudo-Real Rational Maps

TL;DR

This paper studies the structure of the moduli space of complex rational maps, focusing on real and pseudo-real maps, their automorphisms, and their definability over their field of moduli, providing new examples and topological properties.

Contribution

It characterizes the connectedness of real and pseudo-real loci in the moduli space and constructs explicit pseudo-real maps with cyclic automorphism groups, highlighting their non-definability over their field of moduli.

Findings

Both real and pseudo-real loci are connected in the moduli space.

The pseudo-real locus is disconnected.

Explicit examples of pseudo-real maps with cyclic automorphism groups are provided.

Abstract

The moduli space ${\rm M}_{d}$, of complex rational maps of degree $d \geq 2$, is a connected complex orbifold which carries a natural real structure, coming from usual complex conjugation. Its real points are the classes of rational maps admitting antiholomorphic automorphisms. The locus of the real points ${\rm M}_{d}({\mathbb R})$ decomposes as a disjoint union of the loci ${\rm M}_{d}^{\mathbb R}$, consisting of the real rational maps, and ${\mathcal P}_{d}$, consisting of the pseudo-real ones. We obtain that, both ${\rm M}_{d}^{\mathbb R}$ and ${\rm M}_{d}({\mathbb R})$, are connected and that ${\mathcal P}_{d}$ is disconnected. We also observe that the group of holomorphic automorphisms of a pseudo-real rational map is either trivial or a cyclic group. For every $n \geq 1$, we construct pseudo-real rational maps whose group of holomorphic automorphisms is cyclic of order $n$. As the field of moduli of a pseudo-real rational map is contained in ${\mathbb R}$, these maps provide examples of rational maps which are not definable over their field of moduli. It seems that these are the only explicit examples in the literature (Silverman) of rational maps which cannot be defined over their field of moduli. We provide explicit examples of real rational maps which cannot be defined over their field of moduli. Finally, we also observe that every real rational map, which admits a model over the algebraic numbers, can be defined over the real algebraic numbers.