A criterion for the holomorphy of the curvature of smooth planar webs and applications to dual webs of homogeneous foliations on $\mathbb{P}^{2}_{\mathbb{C}}$
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| Format: | Preprint |
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2022
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| _version_ | 1866929255256948736 |
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| author | Bedrouni, Samir Marín, David |
| author_facet | Bedrouni, Samir Marín, David |
| contents | Let $d\geq3$ be an integer. For a holomorphic $d$-web $\mathcal{W}$ on a complex surface $M$, smooth along an irreducible component $D$ of its discriminant $Δ(\mathcal{W}),$ we establish an effective criterion for the holomorphy of the curvature of $\mathcal{W}$ along $D,$ generalizing results on decomposable webs due to Mar\'ın, Pereira and Pirio. As an application, we deduce a complete characterization for the holomorphy of the curvature of the Legendre transform (dual web) $\mathrm{Leg}\mathcal{H}$ of a homogeneous foliation $\mathcal{H}$ of degree $d$ on $\mathbb{P}^{2}_{\mathbb{C}},$ generalizing some of our previous results. This then allows us to study the flatness of the $d$-web $\mathrm{Leg}\mathcal{H}$ in the particular case where the foliation $\mathcal{H}$ is Galois. When the Galois group of $\mathcal{H}$ is cyclic, we show that $\mathrm{Leg}\mathcal{H}$ is flat if and only if $\mathcal{H}$ is given, up to linear conjugation, by one of the two 1-forms $ω_1^{\hspace{0.2mm}d}=y^d\mathrm{d}x-x^d\mathrm{d}y$, $ω_2^{\hspace{0.2mm}d}=x^d\mathrm{d}x-y^d\mathrm{d}y.$ When the Galois group of $\mathcal{H}$ is non-cyclic, we obtain that $\mathrm{Leg}\mathcal{H}$ is always flat. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2209_02381 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A criterion for the holomorphy of the curvature of smooth planar webs and applications to dual webs of homogeneous foliations on $\mathbb{P}^{2}_{\mathbb{C}}$ Bedrouni, Samir Marín, David Dynamical Systems Classical Analysis and ODEs Complex Variables Differential Geometry 14C21, 32S65, 53A60 Let $d\geq3$ be an integer. For a holomorphic $d$-web $\mathcal{W}$ on a complex surface $M$, smooth along an irreducible component $D$ of its discriminant $Δ(\mathcal{W}),$ we establish an effective criterion for the holomorphy of the curvature of $\mathcal{W}$ along $D,$ generalizing results on decomposable webs due to Mar\'ın, Pereira and Pirio. As an application, we deduce a complete characterization for the holomorphy of the curvature of the Legendre transform (dual web) $\mathrm{Leg}\mathcal{H}$ of a homogeneous foliation $\mathcal{H}$ of degree $d$ on $\mathbb{P}^{2}_{\mathbb{C}},$ generalizing some of our previous results. This then allows us to study the flatness of the $d$-web $\mathrm{Leg}\mathcal{H}$ in the particular case where the foliation $\mathcal{H}$ is Galois. When the Galois group of $\mathcal{H}$ is cyclic, we show that $\mathrm{Leg}\mathcal{H}$ is flat if and only if $\mathcal{H}$ is given, up to linear conjugation, by one of the two 1-forms $ω_1^{\hspace{0.2mm}d}=y^d\mathrm{d}x-x^d\mathrm{d}y$, $ω_2^{\hspace{0.2mm}d}=x^d\mathrm{d}x-y^d\mathrm{d}y.$ When the Galois group of $\mathcal{H}$ is non-cyclic, we obtain that $\mathrm{Leg}\mathcal{H}$ is always flat. |
| title | A criterion for the holomorphy of the curvature of smooth planar webs and applications to dual webs of homogeneous foliations on $\mathbb{P}^{2}_{\mathbb{C}}$ |
| topic | Dynamical Systems Classical Analysis and ODEs Complex Variables Differential Geometry 14C21, 32S65, 53A60 |
| url | https://arxiv.org/abs/2209.02381 |