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There is a natural action of the fundamental group
,
, on
.
A positively oriented generator of
operates via the *
monodromy operator* defined by

for
and
.
Let be the decomposition of into the semisimple
part and the unipotent part , and set
.
By the * monodromy theorem* [1,24], the eigenvalues of
are roots of unity and .
Let
be the decomposition of
into the generalized
eigenspaces of ,
,
,
and let
.
For
,
,
is monodromy invariant and defines a holomorphic section in .
The sections
span a
-invariant, finitely
generated, free
-submodule
of rank . Note that
the direct image sheaf
is in general not finitely
generated. The * Gauß-Manin connection* is the * regular*
-module
, the stalk of
at 0 [1,32].

*Christoph Lossen*

*2001-03-21*