Relative Riemann-Hilbert correspondence in dimension one

We prove that, on a Riemann surface, the functor $\mathrm{RH}^S$ constructed in a previous work as a right quasi-inverse of the solution functor from the bounded derived category of regular relative holonomic modules to that of relative constructible complexes satisfies the left quasi-inverse property in a generic sense.Comment: 10 pages. V2: revised version, some mistake corrected, improvement of the presentation. V3: final version to be publishe

tt-Structures for Relative D\mathcal{D}-Modules and tt-Exactness of the de Rham Functor

This paper is a contribution to the study of relative holonomic $\mathcal{D}$-modules. Contrary to the absolute case, the standard $t$-structure on holonomic $\mathcal{D}$-modules is not preserved by duality and hence the solution functor is no longer $t$-exact with respect to the canonical, resp. middle-perverse, $t$-structures. We provide an explicit description of these dual $t$-structures. When the parameter space is 1-dimensional, we use this description to prove that the solution functor as well as the relative Riemann-Hilbert functor are $t$-exact with respect to the dual $t$-structure and to the middle-perverse one while the de Rham functor is $t$-exact for the canonical, resp. middle-perverse, $t$-structures and their duals.Comment: Final version to appear in Journal of Algebr

t-structures for relative D-modules and t-exactness of the de Rham functor

This paper is a contribution to the study of relative holonomic D-modules. Contrary to the absolute case, the standard t-structure on holonomic D-modules is not preserved by duality and hence the solution functor is no longer t-exact with respect to the canonical, resp. middle-perverse, t-structure. We provide an explicit description of these dual t-structures. We use this description to prove that the solution functor as well as the relative Riemann-Hilbert functor are t-exact with respect to the dual t-structure and to the middle-perverse one while the de Rham functor is t-exact for the canonical, resp. middle-perverse, t-structure and their duals

Involutivity of truncated microsupports

Using a result of J-M. Bony, we prove the weak involutivity of truncated microsupports. More precisely, given a sheaf $F$ on a real manifold and an integer $k$, if two functions vanish on the truncated microsupport $Ss_k(F)$, then so does their Poisson bracket.Comment: 9 page

Truncated microsupport and holomorphic solutions of D-modules

We study the truncated microsupport $Ss_k$ of sheaves on a real manifold. Applying our results to the case of $F=RHom_D(M,O)$, the complex of holomorphic solutions of a coherent $D$-module $M$, we show that $Ss_k(F)$ is completely determined by the characteristic variety of $M$. As an application, we obtain an extension theorem for the sections of $H^j(F)$, $j<d$, defined on an open subset whose boundary is non characteristic outside of a complex analytic subvariety of codimension $d$. We also give a characterization of the perversity for ${\bf C}$-constructible sheaves in terms of their truncated microsupports.Comment: 22 page

Regularization of relative holonomic D-modules

Let $X$ and $S$ be complex analytic manifolds where $S$ plays the role of a parameter space. Using the sheaf \DXS^{\infty} of relative differential operators of infinite order, we construct functorially the regular holonomic \DXS-module \shm_{reg} associated to a relative holonomic \DXS-module \shm, extending to the relative case classical theorems by Kashiwara-Kawai: denoting by \shm^{\infty} the tensor product of \shm by \DXS^{\infty} we explicit \shm^{\infty} in terms of the sheaf of holomorphic solutions of \shm and prove that \shm^{\infty} and \shm_{reg}^{\infty} are isomorphic.Comment: This is a revised version following the referee's corrections. The main results are the sam