\Phi_+{\mathcal{O}}(*D)))$. Denote by $\widehat{M}$ the partial Fourier-Laplace transformation with respect to $r$ of $M$, i.e., $\widehat{M}=M$ as ${{\mathds C}}$-vector spaces, and we define an structure of a ${{\mathds C}}[\tau,t]\langle\partial_\tau,\partial_t\rangle$-module on $\widehat{M}$ by $\tau\cdot:=\partial_r$ and $\partial_\tau:=-r\cdot$. Then $\widehat{M}$ is ${{\mathds C}}[\tau,t]\langle\partial_\tau,\partial_t\rangle$-holonomic, with singularities at $\tau=\{0,\infty\}$ and $t=\{0,\infty\}$ at most, regular along $\{t=0\} \cup \{\tau=\infty\}$. Moreover, by putting $\theta=\tau^{-1}$, the localized Fourier-Laplace transformation $\widehat{M}[\tau^{-1}]$ of $M$ is isomorphic to $G(*D)$ as a meromorphic vector bundle with connection (the one on $G(*D)$ being given by formula ).

4.  The restriction $(G_1(*D),\nabla)$ is isomorphic (as a left ${{{{\mathds C}}[t]\langle\partial_t\rangle}}$-module) to ${{\mathbb H}}^0(T, h_+{\mathcal{O}}_V(*D)e^{-f})$, where ${\mathcal{O}}_V(*D)e^{-f}$ is the tensor product of ${\mathcal{O}}_V(*D)$ with a rank one ${\mathcal{O}}_V$-module formally generated by $e^{-f}$, i.e., ${\mathcal{O}}_V(*D)e^{-f}\cong {\mathcal{O}}_V(*D)$ as ${\mathcal{O}}_V$-modules, and the differential of ${\mathcal{O}}_V(*D)e^{-f}$ (i.e., the operator defining its ${\mathcal{D}}_V$-module structure) is given by $d_f:=d-df\wedge$.

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1.  see [@Mebkhout theorème 9.0-8.]

2.  It is well known that for any left ${\mathcal{D}}_V$-module ${\mathcal{M}}$, the direct image complex $\Phi_+ {\mathcal{M}}$ is represented by $$\left({{\mathbb R}}\Phi_*\Omega^{n+\bullet}_V({\mathcal{M}})[\partial_r,\partial_t],d-(df\wedge-\otimes\partial_r)-(dh\wedge-\otimes\partial_t)\right).$$ Putting ${\mathcal{M}}={\mathcal{O}}_V(*D)$ and using that $\Phi$ is affine, we consider the double complex $$E^{p,q}:=\left((\Phi_*\Omega^{p+q}_V(*D)[\partial_r])\otimes\partial_t^q,d-(df\wedge-\otimes\partial_r),-(dh\wedge-\otimes\partial_t )\right)$$ whose total cohomology is ${\mathcal{H}}^{p+q-n}(\Phi_+({\mathcal{O}}_V(*D)))$. The morphism $h$ is smooth restricted to $V\backslash D$, hence, the Koszul complex $(\Omega^\bullet(*D),dh 