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Complex Analytic Desingularization

紀錄類型:	書目-語言資料,印刷品 : Monograph/item
正題名/作者:	Complex Analytic Desingularization/ by José Manuel Aroca, Heisuke Hironaka, José Luis Vicente.
作者:	Aroca, José Manuel.
其他作者:	Hironaka, Heisuke.
面頁冊數:	XXIX, 330 p. 379 illus.online resource. :
Contained By:	Springer Nature eBook
標題:	Algebraic geometry. -
電子資源:	https://doi.org/10.1007/978-4-431-49822-3
ISBN:	9784431498223

Complex Analytic Desingularization
Aroca, José Manuel.

Complex Analytic Desingularization[electronic resource] /by José Manuel Aroca, Heisuke Hironaka, José Luis Vicente. - 1st ed. 2018. - XXIX, 330 p. 379 illus.online resource.

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.

[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.

ISBN: 9784431498223

Standard No.: 10.1007/978-4-431-49822-3doiSubjects--Topical Terms:

1255324
Algebraic geometry.

LC Class. No.: QA564-609

Dewey Class. No.: 516.35

Complex Analytic Desingularization
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