Classification of rational 1-forms on the Riemann sphere up to PSL(2,C)

TL;DR

This paper classifies rational 1-forms on the Riemann sphere with simple poles, studies their symmetries under PSL(2,C), and describes the structure of their moduli spaces and special subfamilies.

Contribution

It introduces a detailed classification framework for rational 1-forms with simple poles, including the structure of isochronous forms and their quotient spaces under PSL(2,C).

Findings

The space of such 1-forms has a complex atlas based on coefficients, zeros, and residues.

Isochronous forms form a real submanifold of the space.

The quotient spaces under PSL(2,C) are stratified by orbit types and explicitly described using invariant functions.

Abstract

We study the family $\Omega^1(-1^s)$ of rational 1--forms on the Riemann sphere, having exactly $-s \leq -2$ simple poles. Three equivalent $(2s-1)$--dimensional complex atlases on $\Omega^1(-1^s)$, using coefficients, zeros--poles and residues--poles of the 1--forms, are recognized. A rational 1--form is isochronous when all their residues are purely imaginary. We prove that the subfamily $\mathcal{RI}\Omega^1(-1^s)$ of isochronous 1--forms is a $(3s-1)$--dimensional real analytic submanifold in the complex manifold $\Omega^1(-1^s)$. The complex Lie group $PSL(2,\mathbb{C})$ acts holomorphically on $\Omega^1(-1^s)$. For $s \geq 3$, the $PSL(2,\mathbb{C})$--action is proper on $\Omega^1(-1^s)$ and $\mathcal{RI}\Omega^1(-1^s)$. Therefore, the quotients $\Omega^1(-1^s)/PSL(2,\mathbb{C})$ and $\mathcal{RI}\Omega^1(-1^s)/PSL(2,\mathbb{C})$ admit a stratification by orbit types. Using an explicit set of $PSL(2,\mathbb{C})$--invariant functions, we give realizations for the quotients $\Omega^1(-1^s)/PSL(2,\mathbb{C})$ and $\mathcal{RI}\Omega^1(-1^s)/PSL(2,\mathbb{C})$.