From the very beginning, algebraic topology has developed under the
influ- ence of the problems posed by trying to understand the
topological properties of complex algebraic varieties (e.g., the
pioneering work by Poincare and Lefschetz). Especially in the work of
Lefschetz [Lf2], the idea is made explicit that singularities are
important in the study of the topology even in the case of smooth
varieties. What is known nowadays about the topology of smooth and
singular vari- eties is quite impressive. The many existing results may
be roughly divided into two classes as follows: (i) very general results
or theories, like stratified Morse theory and (mixed) Hodge theory, see,
for instance, Goresky-MacPherson [GM], Deligne [Del], and Steenbrink
[S6]; and (ii) specific topics of great subtlety and beauty, like the
study of the funda- mental group of the complement in [p>2 of a
singular plane curve initiated by Zariski or Griffiths' theory relating
the rational differential forms to the Hodge filtration on the middle
cohomology group of a smooth projec- tive hypersurface. The aim of this
book is precisely to introduce the reader to some topics in this latter
class. Most of the results to be discussed, as well as the related
notions, are at least two decades old, and specialists use them
intensively and freely in their work. Nevertheless, it is impossible to
find an adequate intro- duction to this subject, which gives a good
feeling for its relations with other parts of algebraic geometry and
topology.
