Classification of rational 1-forms on the Riemann sphere up to $\text {PSL}(2, \mathbb {C})$

Abstract

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