Wallace / Sneddon Homology Theory on Algebraic Varieties

1. Auflage 2014
ISBN: 978-1-4831-5260-8
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 124 Seiten, Web PDF

Reihe: International Series in 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 Poincaré 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 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.

Autoren/Hrsg.

Wallace, Andrew H.

Sneddon, I. N.

Inhaltsverzeichnis

1;Front Cover;1
2;Homology Theory on Algebraic Varieties;4
3;Copyright Page;5
4;Table of Contents;6
5;INTRODUCTION;8
6;CHAPTER I. LINEAR SECTIONS OF AN ALGEBRAIC VARIETY;10
6.1;1. Hyperplane sections of a non-singular variety;10
6.2;2. A family of linear sections of IT;11
6.3;3. The fibring of a variety defined over the complex numbers;16
6.4;4. Homology groups related to V(K);26
7;CHAPTER II. THE SINGULAR SECTIONS;32
7.1;1. Statement of the results;32
7.2;2. Proof of Theorem 11;34
8;CHAPTER III. A PENCIL OF HYPERPLANE SECTIONS;43
8.1;1. The choice of a pencil;43
8.2;2. Notation;46
8.3;3. Reduction to local theorems;47
9;CHAPTER IV. LEFSCHETZ'S FIRST AND SECOND THEOREMS;52
9.1;1. Lefschetz's first main theorem;52
9.2;2. Statement of Lefschetz's second main theorem;58
9.3;3. Sketch proof of Theorem 19;58
9.4;4. Some immediate consequences;63
10;CHAPTER V. PROOF OF LEFSCHETZ'S SECOND THEOREM;65
10.1;1. Deformation theorems;65
10.2;2. Some remarks on Theorem 19;69
10.3;3. Formal verification of Theorem 19; the vanishing cycle;71
10.4;4. Proof of Theorem 19, parts (1) and (2);73
10.5;5. Proof of Theorem 19, part (3);76
11;CHAPTER VI. THE POINCARÉ
FORMULA;81
11.1;1. The automorphisms Ti
;81
11.2;2. Explicit calculation of T;85
11.3;3. The formula T (.)= .– (..d)
d;90
12;CHAPTER VII. THE POINCARÉ
FORMULA; DETAILS OF PROOF;92
12.1;1. Clockwise and anti-clockwise isomorphisms;92
12.2;2. A special representative for T;96
12.3;3. Proof of Theorem 32;97
12.4;4. Proof of Theorem 34;99
13;CHAPTER VIII. INVARIANT CYCLES AND RELATIVE CYCLES;106
13.1;1. Summary of results assumed;106
13.2;2. The intersection and locus operators;107
13.3;3. Direct decomposition for Hr–1(Vo, P)
;110
13.4;4. Direct decomposition of Hr–1(Vo, P)
;111
13.5;5. Proofs of Theorems 41 and 42;115
14;REFERENCES;124