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