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.