A criterion for the holomorphy of the curvature of smooth planar webs and applications to dual webs of homogeneous foliations on P2C
Let d≥3 be an integer. For a holomorphic d-web W on a complex surface M, smooth along an irreducible component D of its discriminant Δ(W), we establish an effective criterion for the holomorphy of the curvature of W along D, generalizing results on decomposable webs due to Marín, Pereira, and Pirio...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2024 |
| País: | España |
| Institución: | Universitat Autònoma de Barcelona |
| Repositorio: | Dipòsit Digital de Documents de la UAB |
| Idioma: | inglés |
| OAI Identifier: | oai:ddd.uab.cat:302280 |
| Acceso en línea: | https://ddd.uab.cat/record/302280 https://dx.doi.org/urn:doi:10.1002/mana.202400150 |
| Access Level: | acceso abierto |
| Palabra clave: | Curvature Galois homogeneous foliation Legendre transform Web |
| Sumario: | Let d≥3 be an integer. For a holomorphic d-web W on a complex surface M, smooth along an irreducible component D of its discriminant Δ(W), we establish an effective criterion for the holomorphy of the curvature of W along D, generalizing results on decomposable webs due to Marín, Pereira, and Pirio. As an application, we deduce a complete characterization for the holomorphy of the curvature of the Legendre transform (dual web) LegH of a homogeneous foliation H of degree d on PC2, generalizing some of our previous results. This then allows us to study the flatness of the d-web LegH in the particular case where the foliation H is Galois. When the Galois group of H is cyclic, we show that LegH is flat if and only if H is given, up to linear conjugation, by one of the two 1-forms ω1d=yddx-xddy, ω2d=xddx-yddy. When the Galois group of H is noncyclic, we obtain that LegH is always flat. |
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