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

  • Alvaro Alvarez-Parrilla EMAIL logo and Jesús Muciño-Raymundo
Published/Copyright: October 5, 2021
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Advances in Geometry
From the journal Advances in Geometry Volume 21 Issue 4

Abstract

We consider the family ℰ (s, r, d) of all singular complex analytic vector fields X(z)=Q(z)P(z)eE(z)∂∂zon 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.

Keywords: Complex analytic vector field; Riemann surface; essential singularity; discrete symmetry group
MSC 2010: 34M35; 32S65; 30D20; 58D19

  1. Communicated by: P. Eberlein

Acknowledgements

The authors wish to thank Adolfo Guillot for useful comments, as well as the anonymous referee for the generous criticism that improved the present work.

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Received: 2019-02-27
Revised: 2020-04-07
Published Online: 2021-10-05
Published in Print: 2021-10-26

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. An 𝔽p2-maximal Wiman sextic and its automorphisms
  3. Pseudo-algebraic Ricci solitons on Einstein nilradicals
  4. The wobbly divisors of the moduli space of rank-2 vector bundles
  5. Symmetries of complex analytic vector fields with an essential singularity on the Riemann sphere
  6. Betti numbers and pseudoeffective cones in 2-Fano varieties
  7. The generating rank of a polar Grassmannian
  8. The Beckman–Quarles theorem via the triangle inequality
  9. On Huisman’s conjectures about unramified real curves
  10. Geodesic orbit Finsler spaces with K ≥ 0 and the (FP) condition
  11. On the Segre invariant for rank two vector bundles on ℙ2
  12. Lifting coarse homotopies
  13. How to construct all metric f-K-contact manifolds
  14. An extremum problem for the power moment of a convex polygon contained in a disc
  15. Sharply transitive sets in PGL2(K)
https://doi.org/10.1515/advgeom-2021-0002
Keywords for this article
Complex analytic vector field; Riemann surface; essential singularity; discrete symmetry group

Articles in the same Issue

  1. Frontmatter
  2. An 𝔽p2-maximal Wiman sextic and its automorphisms
  3. Pseudo-algebraic Ricci solitons on Einstein nilradicals
  4. The wobbly divisors of the moduli space of rank-2 vector bundles
  5. Symmetries of complex analytic vector fields with an essential singularity on the Riemann sphere
  6. Betti numbers and pseudoeffective cones in 2-Fano varieties
  7. The generating rank of a polar Grassmannian
  8. The Beckman–Quarles theorem via the triangle inequality
  9. On Huisman’s conjectures about unramified real curves
  10. Geodesic orbit Finsler spaces with K ≥ 0 and the (FP) condition
  11. On the Segre invariant for rank two vector bundles on ℙ2
  12. Lifting coarse homotopies
  13. How to construct all metric f-K-contact manifolds
  14. An extremum problem for the power moment of a convex polygon contained in a disc
  15. Sharply transitive sets in PGL2(K)
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