Let
M
M
be a left module over a Weyl algebra in characteristic zero. Given natural weight vectors
ν
\nu
and
ω
\omega
, we show that the characteristic varieties arising from filtrations with weight vector
ν
+
s
ω
\nu +s\omega
stabilize to a certain variety determined by
M
M
,
ν
\nu
,
ω
\omega
as soon as the natural number
s
s
grows beyond a bound which depends only on
M
M
and
ν
\nu
but not on
ω
\omega
.
As a consequence, in the notable case when
ν
\nu
is the standard weight vector, these characteristic varieties deform to the critical cone of the
ω
\omega
-characteristic variety of
M
M
as soon as
s
s
grows beyond an invariant of
M
M
.
As an application, we give a new, easy, non-homological proof of a classical result, namely, that the
ω
\omega
-characteristic varieties of
M
M
all have the same Krull dimension.
The set of all
ω
\omega
-characteristic varieties of
M
M
is finite. We provide an upper bound for its cardinality in terms of supports of universal Gröbner bases in the case when
M
M
is cyclic. By the above stability result, we conjecture a second upper bound in terms of total degrees of universal Gröbner bases and of Fibonacci numbers in the case when
M
M
is cyclic over the first Weyl algebra.