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- W3206027987 abstract "We consider the family $mathcal{E}(s,r,d)=Big{ X(z)=frac{Q(z)}{P(z)} e^{E(z)} frac{partial}{partial z} Big},$ with $Q, P, E$ polynomials, deg${Q}=s$, deg$(P)=r$ and deg$(E)=d$, of singular complex analytic vector fields $X$ on the Riemann sphere $widehat{mathbb{C}}$. For $dgeq1$, $Xinmathcal{E}(s,r,d)$ has $s$ zeros and $r$ poles on the complex plane and an essential singularity at infinity. Using the pullback action of the affine group $Aut(mathbb{C})$ and the divisors for $X$, we calculate the isotropy groups $Aut(mathbb{C})_{X}$ and the discrete symmetries for $Xinmathcal{E}(s,r,d)$. Each subfamily $mathcal{E}(s,r,d)_{id}$, of those $X$ with trivial isotropy group in $Aut(mathbb{C})$, is endowed with a holomorphic trivial principal $Aut(mathbb{C})$-bundle structure. Necessary and sufficient conditions in order to ensure the equality $mathcal{E}(s,r,d)=mathcal{E}(s,r,d)_{id}$ and those $Xinmathcal{E}(s,r,d)$ with non-trivial isotropy are realized. Explicit global normal forms for $Xinmathcal{E}(s,r,d)$ are presented. A natural dictionary between vector fields, 1-forms, quadratic differentials and functions is extended to include the presence of non-trivial discrete symmetries $Gamma<Aut(mathbb{C})$." @default.
- W3206027987 created "2021-10-25" @default.
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- W3206027987 date "2021-10-01" @default.
- W3206027987 modified "2023-10-01" @default.
- W3206027987 title "Symmetries of complex analytic vector fields with an essential singularity on the Riemann sphere" @default.
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- W3206027987 doi "https://doi.org/10.1515/advgeom-2021-0002" @default.
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