Tms monograph was originally planned as a series of papers,
the first of which has already appeared, namely [ll]. The
nature of the subject, however, along with the length of the
treatment, made it seem more advisable to rearrange the
work in book form. The material of[ll] appears in a modified
form in Chapters I-IV of this monograph.
The main theorems whose proofs are given here were first
formulated by Lefschetz in [9], and have since turned out to
be of fundamental importance in the topological aspects of
algebraic geometry. These theorems may be briefly described
as follows. Let V be a non-singular r-dimensional algebraic
variety in complex projective space, and let Vo be a nonsingular
hyperplane section of V. Then Lefschetz's first main
theorem states that all cycles of dimension less than r on V
are homologous to cycles on V o' Now V 0 may be taken as a
member of a pencil of hyperplane sections of V, a pencil which
contains only a finite number of singular sections. Lefschetz's
second main theorem, interpreted in terms of relative homology,
shows how to obtain a set of generators for Hr(V, Yo),
one of which is associated in a certain way with each of these
singular sections. The third main result of Lefschetz concerns
the Poincare formula, which describes the variation of cycles
of V 0 as this section is made to vary within a pencil of sections.