AbstractThe Milnor number $$\mu _f$$
μ
f
of a holomorphic function $$f :({\mathbb {C}}^n,0) \rightarrow ({\mathbb {C}},0)$$
f
:
(
C
n
,
0
)
→
(
C
,
0
)
with an isolated singularity has several different characterizations as, for example: 1) the number of critical points in a morsification of f, 2) the middle Betti number of its Milnor fiber $$M_f$$
M
f
, 3) the degree of the differential $${\text {d}}f$$
d
f
at the origin, and 4) the length of an analytic algebra due to Milnor’s formula $$\mu _f = \dim _{\mathbb {C}}{\mathcal {O}}_n/{\text {Jac}}(f)$$
μ
f
=
dim
C
O
n
/
Jac
(
f
)
. Let $$(X,0) \subset ({\mathbb {C}}^n,0)$$
(
X
,
0
)
⊂
(
C
n
,
0
)
be an arbitrarily singular reduced analytic space, endowed with its canonical Whitney stratification and let $$f :({\mathbb {C}}^n,0) \rightarrow ({\mathbb {C}},0)$$
f
:
(
C
n
,
0
)
→
(
C
,
0
)
be a holomorphic function whose restriction f|(X, 0) has an isolated singularity in the stratified sense. For each stratum $${\mathscr {S}}_\alpha $$
S
α
let $$\mu _f(\alpha ;X,0)$$
μ
f
(
α
;
X
,
0
)
be the number of critical points on $${\mathscr {S}}_\alpha $$
S
α
in a morsification of f|(X, 0). We show that the numbers $$\mu _f(\alpha ;X,0)$$
μ
f
(
α
;
X
,
0
)
generalize the classical Milnor number in all of the four characterizations above. To this end, we describe a homology decomposition of the Milnor fiber $$M_{f|(X,0)}$$
M
f
|
(
X
,
0
)
in terms of the $$\mu _f(\alpha ;X,0)$$
μ
f
(
α
;
X
,
0
)
and introduce a new homological index which computes these numbers directly as a holomorphic Euler characteristic. We furthermore give an algorithm for this computation when the closure of the stratum is a hypersurface.
A generalization of Milnor’s formula
