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Closed Meromorphic 1-Forms

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Handbook of Geometry and Topology of Singularities V: Foliations

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Abstract

We review properties of closed meromorphic 1-forms and of the foliations defined by them. We present and explain classical results from foliation theory, like index theorems, existence of separatrices, and resolution of singularities under the lenses of the theory of closed meromorphic 1-forms and flat meromorphic connections. We apply the theory to investigate the algebraicity of separatrices in a semi-global setting (neighborhood of a compact curve contained in the singular set of the foliation), and the geometry of smooth hypersurfaces with numerically trivial normal bundle on compact Kähler manifolds.

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Acknowledgements

This text grew out from notes for a course taught at IMPA during the Southern Hemisphere Summer of 2022 and was prepared to answer an invitation of Felipe Cano and Pepe Seade (made on June 2021) to write a survey on a foliation theoretic topic of my choice. It is a pleasure to thank them for giving me the opportunity, and providing the impetus, to think and share my thoughts about closed meromorphic 1-forms and the foliations defined by them. I also thank Andreas Höring for useful correspondence about Stein complements of hypersurfaces on compact Kähler manifolds. I am also indebted to Maycol Falla Luza, Thiago Fassarella, Frédéric Touzet, and Sebastián Velazquez for reading parts of preliminary versions of this work, catching quite a few misprints, and suggesting a number of improvements. A preliminary version of this text ended up at the hands of a thorough and thoughtful referee. Their feedback saved me from a number of embarrassing mistakes and allowed me to improve the exposition.

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  1. IMPA, Rio de Janeiro, Brazil

    Jorge Vitório Pereira

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  1. Jorge Vitório Pereira
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  1. Departamento de Algebra, Análisis Matemático, Geometría y Topología, Universidad de Valladolid, Valladolid, Spain

    Felipe Cano

  2. Instituto de Matemáticas, Unidad Cuernavaca, Universidad Nacional Autónoma de México, Cuernavaca, Mexico City, Mexico

    José Luis Cisneros-Molina

  3. Centre de mathématiques et informatique, Université d’Aix-Marseille, Marseille, France

    Lê Dũng Tráng

  4. Instituto de Matemáticas, Unidad Cuernavaca, Universidad Nacional Autónoma de México, Cuernavaca, Mexico City, Mexico

    José Seade

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Pereira, J.V. (2024). Closed Meromorphic 1-Forms. In: Cano, F., Cisneros-Molina, J.L., Dũng Tráng, L., Seade, J. (eds) Handbook of Geometry and Topology of Singularities V: Foliations. Springer, Cham. https://doi.org/10.1007/978-3-031-52481-3_9

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