Symmetries of complex analytic vector fields with an essential singularity on the Riemann sphere

Abstract

Abstract We consider the family ℰ (s, r, d) of all singular complex analytic vector fields X ( z ) = Q ( z ) P ( z ) e E ( z ) ∂ ∂ z $X(z)=\frac{Q(z)}{P(z)}{{e}^{E(z)}}\frac{\partial }{\partial z}$ on the Riemann sphere where Q, P, ℰ are polynomials with deg Q = s, deg P = r and deg ℰ = d ≥ 1. Using the pullback action of the affine group Aut(ℂ) and the divisors for X, we calculate the isotropy groups Aut(ℂ) X of discrete symmetries for X ∈ ℰ (s, r, d). The subfamily ℰ (s, r, d)id of those X with trivial isotropy group in Aut(ℂ) is endowed with a holomorphic trivial principal Aut(ℂ)-bundle structure. A necessary and sufficient arithmetic condition on s, r, d ensuring the equality ℰ (s, r, d) = ℰ (s, r, d)id is presented. Moreover, those X ∈ ℰ (s, r, d) \ ℰ (s, r, d)id with non-trivial isotropy are realized. This yields explicit global normal forms for all X ∈ ℰ (s, r, d). A natural dictionary between analytic tensors, vector fields, 1-forms, orientable quadratic differentials and functions on Riemann surfaces M is extended as follows. In the presence of nontrivial discrete symmetries Γ < Aut(M), the dictionary describes the correspondence between Γ-invariant tensors on M and tensors on M /Γ.