$b(s)\in{{\mathds C}}[s]$ such that for all $k\in{{\mathds Z}}$, we have $b(-\partial_t t + k)U_k{\mathcal{M}}\subset U_{k-1}{\mathcal{M}}$.

4.  A coherent ${\mathcal{D}}_X$-module ${\mathcal{M}}$ is called specializable iff locally there exists a good $V$-filtration $U_\bullet{\mathcal{M}}$ having a Bernstein polynomial. Equivalently, for any local section $m\in {\mathcal{M}}$ there is a non-zero polynomial $b_m(s)$ (the Bernstein polynomial of $m$) such that $b_m(-\partial_t t)\,m\in V_{-1}{\mathcal{D}}_X\cdot m$.

5.  A holonomic ${\mathcal{D}}_X$-module is specializable along any smooth hypersurface $Y$.

The following evident corollary gives an example of a $V$-filtration that will be used later.

Consider the left ${{{{\mathds C}}[t]\langle\partial_t\rangle}}$-module $G_1(*D)$ from above. Then $$U_k G_1(*D):=V_k {{{{\mathds C}}[t]\langle\partial_t\rangle}}\cdot G_1(\log\,D)$$ defines a good V-filtration on $G_1(*D)$, whose Bernstein polynomial is exactly $b_{G_1(\log\,D)}(s)$. Moreover, we have $$U_0 G_1(*D) = G_1(\log\,D) = V_0 {{{{\mathds C}}[t]\langle\partial_t\rangle}}\cdot \omega_1.$$

We will also use $V$-filtrations for ${\mathcal{D}}_{T\times V}$-modules. The following result is well known, see, e.g., [@Mal6], [@MebkhMais lemme 4.4-1].

\[lem:BernsteinVFilt\]

1.  Let $h\in {\mathcal{O}}_V$ an arbitrary function, seen as a morphism $h:V\rightarrow T$. Denote by $i_h:V\hookrightarrow T\times V$ the graph embedding, with image $\Gamma_h$. Put ${\mathcal{N}}:=(i_h)_+{\mathcal{O}}_V$, then ${\mathcal{N}}\cong {\mathcal{O}}_V[\partial_t] \cong {\mathcal{O}}_{T\times V}(*\Gamma_h)/{\mathcal{O}}_{T\times V}\cong {\mathcal{D}}_{T\times V}\delta(t-h)$. A good $V$-filtration with respect to the hypersurface $\{0\}\times V$ on ${\mathcal{N}}$ is defined by putting, for all $k \in {{\mathds 