The aim of the present article is to construct analytic invariants for a germ of a holomorphic function having a one-dimensional critical locus
S
S
. This is done for a large class of such germs containing for instance any quasi-homogeneous germ at the origin. More precisely, aside from the Brieskorn
(
a
,
b
)
(a,b)
-module at the origin and a (locally constant along
S
∗
:=
S
∖
{
0
}
S^* : = S \setminus \{0\}
) sheaf
H
n
\mathcal {H}^n
of
(
a
,
b
)
(a,b)
-modules associated with the transversal hypersurface singularities along each connected component of
S
∗
S^*
, we construct also
(
a
,
b
)
(a,b)
-modules “with supports”
E
c
E_c
and
E
c
∩
S
′
E’_{c \cap \, S}
.
An interesting consequence of the local study along
S
∗
S^*
is the corollary showing that for a germ with an isolated singularity, the largest sub-
(
a
,
b
)
(a,b)
-module having a simple pole in its Brieskorn-
(
a
,
b
)
(a,b)
-module is independent of the choice of a reduced equation for the corresponding hypersurface germ.
We also give precise relations between these various
(
a
,
b
)
(a,b)
-modules via an exact commutative diagram. This is an
(
a
,
b
)
(a,b)
-linear version of the tangling phenomenon for consecutive strata we have previously studied in the “topological” setting for the localized Gauss-Manin system of
f
f
.
Finally we show that in our situation there exists a non-degenerate
(
a
,
b
)
(a,b)
-sesquilinear pairing
\[
h
:
E
×
E
c
∩
S
′
⟶
|
Ξ
′
|
2
h : E \times E’_{c\,\cap \, S} \longrightarrow \vert \Xi ’ \vert ^2
\]
where
|
Ξ
′
|
2
\vert \Xi ’ \vert ^2
is the space of formal asymptotic expansions at the origin for fiber integrals. This generalizes the canonical hermitian form defined in 1985 for the isolated singularity case (for the
(
a
,
b
)
(a,b)
-module version see the recent 2005 paper). Its topological analogue (for the eigenvalue
1
1
of the monodromy) is the non-degenerate sesquilinear pairing
\[
h
:
H
c
∩
S
n
(
F
,
C
)
=
1
×
H
n
(
F
,
C
)
=
1
→
C
h : H^n_{c\,\cap \,S}(F, \mathbb {C})_{=1} \times H^n(F, \mathbb {C})_{=1} \to \mathbb {C}
\]
defined in an earlier paper for an arbitrary germ with a one-dimensional critical locus. Then we show this sesquilinear pairing is related to the non-degenerate sesquilinear pairing introduced on the sheaf
H
n
\mathcal {H}^n
via the canonical Hermitian form of the transversal hypersurface singularities.