Homology Theory on Algebraic Varieties

Andrew H. Wallace

• Publisher: Dover Publications
• Year: 2014
• Print ISBN: 9780486787848
• eBook ISBN: 9780486799902
• Topic: Mathematics
• Subtopic: Abstract Algebra
• Index: Mathematics

Concise and authoritative, this monograph is geared toward advanced undergraduate and graduate students. The main theorems whose proofs are given here were first formulated by Lefschetz and have since turned out to be of fundamental importance in the topological aspects of algebraic geometry. The proofs are fairly elaborate and involve a considerable amount of detail; therefore, some appear in separate chapters that include geometrical descriptions and diagrams.
The treatment begins with a brief introduction and considerations of linear sections of an algebraic variety as well as singular and hyperplane sections. Subsequent chapters explore Lefschetz's first and second theorems with proof of the second theorem, the Poincaré formula and details of its proof, and invariant and relative cycles.

CHAPTER V

PROOF OF LEFSCHETZ′S SECOND THEOREM

1.Deformation theorems

It is clear that the method sketched in the last chapter for the proof of Theorem 19 depends on some mechanism which will provide the necessary deformations and shrinkings, a mechanism which, in addition, allows these operations to be carried out in an arbitrary neighbourhood of C′, in the notation already introduced in §3, Chapter III. In this section two theorems will be proved which will enable the details of the proof sketched in §3, Chapter IV to be carried out.

The two arcs λ₁ and λ₂ in the complex w-plane have already been introduced in §3, Chapter IV; they are the arcs traced out by the two values of w(z) as z traces the arc λ in the z-plane. Also if t is the parameter on λ then λ₁

λ₂ = λ′ can be parametrized by s where s² − t. For the point z on λ with parameter t the two corresponding values of w(z), namely w₁(z) and w₂(z), have parameter values

and

on λ′.

Consider now the following set ∑ a subset of λ × λ′, which in its turn is a subset, namely a 2-cell, of the (z, w)-plane. ∑ is to consist of the points (z, w) where z ∈ λ, of parameter t, say, and w ∈ λ′ of parameter s such that

. It is not hard to see that ∑ is a 2-cell which can be assumed, by taking λ small enough, to be contained in a neighbourhood of (z′, w′), a neighbourhood which can be assumed to be a 4-cell, and that the boundary of ∑ is formed by th...

Table of contents

• Cover
• Title Page
• Copyright Page
• Contents
• Introduction
• I. Linear Sections of an Algebraic Variety
• II. The Singular Sections
• III. A Pencil of Hyperplane Sections
• IV. Lefschetz’s First and Second Theorems
• V. Proof of Lefschetz’s Second Theorem
• VI. The Poincaré Formula
• VII. The Poincaré Formula; Details of Proof
• VIII. Invariant Cycles and Relative Cycles
• References