Homology Theory on Algebraic Varieties (eBook)
124 Seiten
Elsevier Science (Verlag)
978-1-4831-5260-8 (ISBN)
This book is comprised of eight chapters and begins with a discussion on linear sections of an algebraic variety, with emphasis on the fibring of a variety defined over the complex numbers. The next two chapters focus on singular sections and hyperplane sections, focusing on the choice of a pencil in the latter case. The reader is then introduced to Lefschetz's first and second theorems, together with their corresponding proofs. The Poincaré formula and its proof are also presented, with particular reference to clockwise and anti-clockwise isomorphisms. The final chapter is devoted to invariant cycles and relative cycles.
This volume will be of interest to students, teachers, and practitioners of pure and applied mathematics.
Homology Theory on Algebraic Varieties, Volume 6 deals with the principles of homology theory in algebraic geometry and includes the main theorems first formulated by Lefschetz, one of which is interpreted in terms of relative homology and another concerns the Poincare formula. The actual details of the proofs of these theorems are introduced by geometrical descriptions, sometimes aided with diagrams. This book is comprised of eight chapters and begins with a discussion on linear sections of an algebraic variety, with emphasis on the fibring of a variety defined over the complex numbers. The next two chapters focus on singular sections and hyperplane sections, focusing on the choice of a pencil in the latter case. The reader is then introduced to Lefschetz's first and second theorems, together with their corresponding proofs. The Poincare formula and its proof are also presented, with particular reference to clockwise and anti-clockwise isomorphisms. The final chapter is devoted to invariant cycles and relative cycles. This volume will be of interest to students, teachers, and practitioners of pure and applied mathematics.
Front Cover 1
Homology Theory on Algebraic Varieties 4
Copyright Page 5
Table of Contents 6
INTRODUCTION 8
CHAPTER I. LINEAR SECTIONS OF AN ALGEBRAIC VARIETY 10
1. Hyperplane sections of a non-singular variety 10
2. A family of linear sections of IT 11
3. The fibring of a variety defined over the complex numbers 16
4. Homology groups related to V(K) 26
CHAPTER II. THE SINGULAR SECTIONS 32
1. Statement of the results 32
2. Proof of Theorem 11 34
CHAPTER III. A PENCIL OF HYPERPLANE SECTIONS 43
1. The choice of a pencil 43
2. Notation 46
3. Reduction to local theorems 47
CHAPTER IV. LEFSCHETZ'S FIRST AND SECOND THEOREMS 52
1. Lefschetz's first main theorem 52
2. Statement of Lefschetz's second main theorem 58
3. Sketch proof of Theorem 19 58
4. Some immediate consequences 63
CHAPTER V. PROOF OF LEFSCHETZ'S SECOND THEOREM 65
1. Deformation theorems 65
2. Some remarks on Theorem 19 69
3. Formal verification of Theorem 19 the vanishing cycle
4. Proof of Theorem 19, parts (1) and (2) 73
5. Proof of Theorem 19, part (3) 76
CHAPTER VI. THE POINCARÉ
81
1. The automorphisms Ti
81
2. Explicit calculation of T 85
3. The formula T (.)= .– (..d)
90
CHAPTER VII. THE POINCARÉ
DETAILS OF PROOF 92
1. Clockwise and anti-clockwise isomorphisms 92
2. A special representative for T 96
3. Proof of Theorem 32 97
4. Proof of Theorem 34 99
CHAPTER VIII. INVARIANT CYCLES AND RELATIVE CYCLES 106
1. Summary of results assumed 106
2. The intersection and locus operators 107
3. Direct decomposition for Hr–1(Vo, P)
110
4. Direct decomposition of Hr–1(Vo, P)
111
5. Proofs of Theorems 41 and 42 115
REFERENCES 124
Erscheint lt. Verlag | 10.7.2014 |
---|---|
Sprache | englisch |
Themenwelt | Mathematik / Informatik ► Mathematik ► Allgemeines / Lexika |
Technik | |
ISBN-10 | 1-4831-5260-X / 148315260X |
ISBN-13 | 978-1-4831-5260-8 / 9781483152608 |
Haben Sie eine Frage zum Produkt? |
![PDF](/img/icon_pdf_big.jpg)
Größe: 12,5 MB
Kopierschutz: Adobe-DRM
Adobe-DRM ist ein Kopierschutz, der das eBook vor Mißbrauch schützen soll. Dabei wird das eBook bereits beim Download auf Ihre persönliche Adobe-ID autorisiert. Lesen können Sie das eBook dann nur auf den Geräten, welche ebenfalls auf Ihre Adobe-ID registriert sind.
Details zum Adobe-DRM
Dateiformat: PDF (Portable Document Format)
Mit einem festen Seitenlayout eignet sich die PDF besonders für Fachbücher mit Spalten, Tabellen und Abbildungen. Eine PDF kann auf fast allen Geräten angezeigt werden, ist aber für kleine Displays (Smartphone, eReader) nur eingeschränkt geeignet.
Systemvoraussetzungen:
PC/Mac: Mit einem PC oder Mac können Sie dieses eBook lesen. Sie benötigen eine
eReader: Dieses eBook kann mit (fast) allen eBook-Readern gelesen werden. Mit dem amazon-Kindle ist es aber nicht kompatibel.
Smartphone/Tablet: Egal ob Apple oder Android, dieses eBook können Sie lesen. Sie benötigen eine
Geräteliste und zusätzliche Hinweise
Buying eBooks from abroad
For tax law reasons we can sell eBooks just within Germany and Switzerland. Regrettably we cannot fulfill eBook-orders from other countries.
aus dem Bereich