The Milnor number is a powerful invariant of an isolated,
complex, affine hypersurface singularity. It provides data about the local,
ambient, topological-type of the hypersurface, and the constancy of the Milnor
number throughout a family implies that Thom's $a_f$ condition holds
and that the local, ambient, topological-type is constant in the family. Much
of the usefulness of the Milnor number is due to the fact that it can be
effectively calculated in an algebraic manner.
The Lê cycles and numbers are a generalization of the
Milnor number to the setting of complex, affine hypersurface singularities,
where the singular set is allowed to be of arbitrary dimension. As with the
Milnor number, the Lê numbers provide data about the local, ambient,
topological-type of the hypersurface, and the constancy of the Lê numbers
throughout a family implies that Thom's $a_f$ condition holds and that
the Milnor fibrations are constant throughout the family. Again, much of the
usefulness of the Lê numbers is due to the fact that they can be
effectively calculated in an algebraic manner.
In this work, we generalize the Lê cycles and numbers to
the case of hypersurfaces inside arbitrary analytic spaces. We define the
Lê-Vogel cycles and numbers, and prove that the Lê-Vogel numbers
control Thom's $a_f$ condition. We also prove a relationship between
the Euler characteristic of the Milnor fibre and the Lê-Vogel numbers.
Moreover, we give examples which show that the Lê-Vogel numbers are
effectively calculable.
In order to define the Lê-Vogel cycles and numbers, we
require, and include, a great deal of background material on Vogel cycles,
analytic intersection theory, and the derived category. Also, to serve as a
model case for the Lê-Vogel cycles, we recall our earlier work on the
Lê cycles of an affine hypersurface singularity.
Readership
Graduate students and research mathematicians interested in several
complex variables and analytic spaces.