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An Invitation to Web Geometry

  • Book
  • © 2015

Overview

Authors:
  1. Jorge Vitório Pereira

    1. Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brazil

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  2. Luc Pirio

    1. UMR 6625 – CNRS, Institut de Recherches Mathématiques de Rennes, Rennes, France

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    You can also search for this author in PubMed   Google Scholar

  • I?ncludes a short survey of the history of the field
  • Presentation is elementary and clear
  • Allows the reader to have a global picture of what were and what are the main questions of the field

Part of the book series: IMPA Monographs (IMPA, volume 2)

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Table of contents (6 chapters)

  1. Front Matter

    Pages i-xvii
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  2. Local and Global Webs

    • Jorge Vitório Pereira, Luc Pirio
    Pages 1-37

  3. Abelian Relations

    • Jorge Vitório Pereira, Luc Pirio
    Pages 39-64

  4. Abel’s Addition Theorem

    • Jorge Vitório Pereira, Luc Pirio
    Pages 65-90

  5. The Converse to Abel’s Theorem

    • Jorge Vitório Pereira, Luc Pirio
    Pages 91-114

  6. Algebraization of Maximal Rank Webs

    • Jorge Vitório Pereira, Luc Pirio
    Pages 115-149

  7. Exceptional Webs

    • Jorge Vitório Pereira, Luc Pirio
    Pages 151-194

  8. Back Matter

    Pages 195-213
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Keywords

About this book

This book takes an in-depth look at abelian relations of codimension one webs in the complex analytic setting. In its classical form, web geometry consists in the study of webs up to local diffeomorphisms. A significant part of the theory revolves around the concept of abelian relation, a particular kind of functional relation among the first integrals of the foliations of a web. Two main focuses of the book include how many abelian relations can a web carry and which webs are carrying the maximal possible number of abelian relations. The book offers complete proofs of both Chern’s bound and Trépreau’s algebraization theorem, including all the necessary prerequisites that go beyond elementary complex analysis or basic algebraic geometry. Most of the examples known up to date of non-algebraizable planar webs of maximal rank are discussed in detail. A historical account of the algebraization problem for maximal rank webs of codimension one is also presented.

Reviews

“This book gives an important contribution on the study of web geometry and its relation with algebraic and complex geometry. … We also note that the book is presented in a self-contained way. … We remark that several very interesting and different examples are presented and the book moreover illustrates the interplay with several areas of mathematics.” (Arturo Fernández-Pérez, Mathematical Reviews, May, 2016)

“The main aim of the book under review is to present the basic results on this fascinating area of geometry. … The book is written in a clear and precise style. … this monograph will be of great interest to graduate students and researchers working in the field of web geometry.” (Gabriel Eduard Vilcu, zbMATH 1321.53003, 2015)

Authors and Affiliations

  • Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brazil

    Jorge Vitório Pereira

  • UMR 6625 – CNRS, Institut de Recherches Mathématiques de Rennes, Rennes, France

    Luc Pirio

About the authors

Jorge Vitorio Pereira is a Research Associate at IMPA (Instituto Nacional de Matematica Pura e Aplicada). Luc Pirio leads research efforts at CNRS.

Bibliographic Information

  • Book Title: An Invitation to Web Geometry

  • Authors: Jorge Vitório Pereira, Luc Pirio

  • Series Title: IMPA Monographs

  • DOI: https://doi.org/10.1007/978-3-319-14562-4

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2015

  • Hardcover ISBN: 978-3-319-14561-7Published: 23 March 2015

  • Softcover ISBN: 978-3-319-38508-2Published: 09 October 2016

  • eBook ISBN: 978-3-319-14562-4Published: 23 February 2015

  • Edition Number: 1

  • Number of Pages: XVII, 213

  • Number of Illustrations: 12 b/w illustrations, 17 illustrations in colour

  • Topics: Algebraic Geometry, Differential Geometry, Several Complex Variables and Analytic Spaces

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