Higher Bernstein Polynomials and Multiple Poles of $\frac{1}{\Gamma(\lambda)} \int_X \vert f \vert^{2\lambda} \bar f^{-h}\rho\omega\wedge \bar \omega'$

The goal of this paper is to give a converse to the main result of my previous paper \cite{[B.22]}, so to prove the existence of a pole with an hypothesis on the Bernstein polynomial of the $(a,b)$-module generated by the germ $\omega \in \Omega^{n+1}_0$. A difficulty to prove such a result comes from the use of the formal completion in $f$ of the Brieskorn module of the holomorphic germ $f: (\mathbb{C}^{n+1}, 0) \to (\mathbb{C}, 0)$ which does not give access to the cohomology of the Milnor's fiber of $f$, which by definition, is outside $\{f=0\}$. This leads to introduce convergent $(a,b)$-modules which allow this passage. In order to take in account Jordan blocs of the monodromy in our result we introduce the semi-simple filtration of a (convergent) geometric $(a,b)$-module and define the higher order Bernstein polynomials in this context which corresponds to a decomposition of the ''standard'' Bernstein polynomial in the case of frescos. Our main result is to show that the existence of a root in $-\alpha - \mathbb{N}$ for the $p$-th Bernstein polynomial of the fresco generated by a holomorphic form $\omega \in \Omega^{n+1}_0$ in the (convergent) Brieskorn $(a,b)$-module $H^{n+1}_0 $ associated to $f$, under the hypothesis that $f$ has an isolated singularity at the origin relative to the eigenvalue $\exp(2i\pi\alpha)$ of the monodromy, produces poles of order at least $p$ for the meromorphic extension of the (conjugate) analytic functional, for some $h \in \mathbb{Z}$: $$\omega' \in \Omega^{n+1}_0 \mapsto \frac{1}{\Gamma(\lambda)}\int_{\mathbb{C}^{n+1}} \vert f\vert^{2\lambda} \bar f^{-h} \rho\omega\wedge \bar \omega' $$ at points $-\alpha - N$ for $N$ and $h$ well chosen integers. This result is new, even for $p = 1$. As a corollary, this implies that in this situation the existence of a root in $-\alpha -\mathbb{N}$ for the $p$-th Bernstein polynomial of the fresco generated by a holomorphic form $\omega \in \Omega^{n+1}_0$ implies the existence of at least $p$ roots (counting multiplicities) for the usual reduced Bernstein polynomial of the germ $(f, 0)$. In the case of an isolated singularity we obtain that for each $\alpha \in ]0, 1] \cap \mathbb{Q}$ the biggest root $-\alpha - m$ of the reduced Bernstein polynomial of $f$ in $-\alpha - \mathbb{N}$ produces a pole at $-\alpha - m$ for some $h \in \mathbb{Z}$ for the meromorphic extension of the distribution $$\square \longrightarrow \frac{1}{\Gamma(\lambda)}\vert f\vert^{2\lambda} \bar f^{-h}\square.$$