
Complex Analytic Desingularization
Springer (Publisher)
1st Edition
Published on 14. November 2018
Book
Hardback
360 pages
978-4-431-70218-4 (ISBN)
Description
[From the foreword by B. Teissier] The main ideas of the proof of resolution of singularities of complex-analytic spaces presented here were developed by Heisuke Hironaka in the late 1960s and early 1970s. Since then, a number of proofs, all inspired by Hironaka's general approach, have appeared, the validity of some of them extending beyond the complex analytic case. The proof has now been so streamlined that, although it was seen 50 years ago as one of the most difficult proofs produced by mathematics, it can now be the subject of an advanced university course. Yet, far from being of historical interest only, this long-awaited book will be very rewarding for any mathematician interested in singularity theory. Rather than a proof of a canonical or algorithmic resolution of singularities, what is presented is in fact a masterly study of the infinitely near "worst" singular points of a complex analytic space obtained by successive "permissible" blowing ups and of the way to tame them using certain subspaces of the ambient space. This taming proves by an induction on the dimension that there exist finite sequences of permissible blowing ups at the end of which the worst infinitely near points have disappeared, and this is essentially enough to obtain resolution of singularities. Hironaka's ideas for resolution of singularities appear here in a purified and geometric form, in part because of the need to overcome the globalization problems appearing in complex analytic geometry.
In addition, the book contains an elegant presentation of all the prerequisites of complex analytic geometry, including basic definitions and theorems needed to follow the development of ideas and proofs. Its epilogue presents the use of similar ideas in the resolution of singularities of complex analytic foliations. This text will be particularly useful and interesting for readers of the younger generation who wish to understand one of the most fundamental results in algebraic and analytic geometry and invent possible extensions and applications of the methods created to prove it.
More details
Product info
HC runder Rücken kaschiert
Edition
1st ed. 2018
Language
English
Place of publication
Tokyo
Japan
Target group
College/higher education
Professional and scholarly
Illustrations
379 s/w Abbildungen
379 Illustrations, black and white; XXIX, 330 p. 379 illus.
Dimensions
Height: 241 mm
Width: 160 mm
Thickness: 25 mm
Weight
705 gr
ISBN-13
978-4-431-70218-4 (9784431702184)
DOI
10.1007/978-4-431-49822-3
Schweitzer Classification
Other editions
Additional editions

José Manuel Aroca | Heisuke Hironaka | José Luis Vicente
Complex Analytic Desingularization
E-Book
11/2018
1st Edition
Springer
€149.79
Available for download
Content
Prologue.- 1 Complex-Analytic Spaces and Elements.- 2 The Weierstrass Preparation Theorem and Its Consequences.- 3 Maximal Contact.- 4 Groves and Polygroves.- 5 The Induction Process.- Epilogue: Singularities of differential equations.- Bibliography.- Index.