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

We study the family Ω 1 ( - 1 s ) of rational 1-forms on the Riemann sphere, having exactly - s ≤ - 2 simple poles. Three equivalent ( 2 s - 1 ) -dimensional complex atlases on Ω 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 RI Ω 1 ( - 1 s ) of isochronous 1-forms is a ( 3 s - 1 ) -dimensional real analytic submanifold in the complex manifold Ω 1 ( - 1 s ) . The complex Lie group PSL ( 2 , C ) acts holomorphically on Ω 1 ( - 1 s ) . For s ≥ 3 , the PSL ( 2 , C ) -action is proper on Ω 1 ( - 1 s ) and RI Ω 1 ( - 1 s ) . Therefore, the quotients Ω 1 ( - 1 s ) / PSL ( 2 , C ) and RI Ω 1 ( - 1 s ) / PSL ( 2 , C ) admit a stratification by orbit types. Realizations for the quotients Ω 1 ( - 1 s ) / PSL ( 2 , C ) and RI Ω 1 ( - 1 s ) / PSL ( 2 , C ) are given, using an explicit set of PSL ( 2 , C ) -invariant functions.