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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.
List of contents
Local and Global Webs.- Abelian Relations.- Abel's Addition Theorem.- The Converse to Abel's Theorem.- Algebraization.- Exceptional Webs.
About the author
Jorge Vitorio Pereira is a Research Associate at IMPA (Instituto Nacional de Matematica Pura e Aplicada). Luc Pirio leads research efforts at CNRS.
Summary
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.
Additional text
“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)
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"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)
