The Milnor number of an affine complex analytic hypersurface with an isolated singularity
has been a ridiculously successful invariant: it can be effectively calculated, it determines
the homotopy-type of the Milnor fibre, and its constancy in a family controls much of the
geometry and topology of the family.
It is no wonder that there have been myriad attempts to generalize the Milnor number
to the cases where the singularity is non-isolated or where the underlying space is arbitrary.
From the topological side, one might suspect that the Betti numbers or the Euler charac-
teristic of the Milnor fibre might be reasonable substitutes for the Milnor number. From
a differential geometry point-of-view, one can consider various notions of indices of vector
fields. From the algebraic side, there are sheaf-theoretic generalizations of the Milnor num-
ber. A nice, but by no means complete, expository discussion of the Milnor number and its
generalizations is contained in [Tel].
Our own work on generalizing Milnor numbers began with the Le varieties, Le cycles,
and Le numbers of a non-isolated affine hypersurface singularity; this work appeared in
[Mas6], [Mas8], [Mas9], and [Masl4]. If U is an open subset of C
n + 1
, / : U - C is an
analytic function, and z := (ZQ, • • • , zn) is a linear choice of coordinates for C
n + 1
, then the
Le numbers, A?
, . . . , A^z, have a number of very desirable properties.
Let s denote the dimension of the critical locus of / at a point p G / _ 1 (0) . Then, the
Le numbers, A* , are zero for i s, and if s — 0, then A° is precisely the Milnor number.
More generally, all of the Le numbers are effectively calculable, and the Milnor fibre has a
handle decomposition in which the number of handles attached of a given index is given by
the corresponding Le number. The constancy of the Le numbers in a family implies that
Thorn's af holds for the total space (the union of the members of the family) and that the
Milnor fibrations are constant in the family. All of these properties, and more, are proved
in Part II of this book.
There is only one question which is addressed by the results of this book: how does
one generalize the Le numbers of an analytic function to the setting where
the underlying space is no longer affine, but, rather, is an arbitrarily singular
Obviously, to answer this, we need to consider how the Le numbers are defined. The Le
numbers are intersection numbers of the Le cycles with affine linear subspaces defined by
the coordinate choice z. Hence, if we could generalize the Le cycles, A^z, then we would
know how to generalize the Le numbers.
The Le cycles are defined by looking at the relative polar varieties ([L-T2], [Te3], and
[Te4]) of / , with the correct cycle structure, and using them to give a "decomposition" of the
Jacobian ideal of / into a collection of cycles. In a more general setting, this decomposition
has been studied by Vogel [Vo] and van Gastel [Gasl], [Gas2]. The first problem, when
the underlying space is arbitrary, is that there are three competing definitions for the polar
cycles - all of which involve gap sheaves, a notion first studied in [Si-TV].
Part I of this book discusses these three competing definitions in the general context of
decomposing any ideal, not necessarily one related to the Jacobian of a function. In Part I,
we prove that the Le-Iomdine formulas, which are such an important result on Le numbers,