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Ana Rita Martins

Publications and source records attributed to Ana Rita Martins.

6 recordsLinked to original sources
arXiv2013

Extension of functors for algebras of formal deformation

Suppose we are given complex manifolds $X$ and $Y$ together with substacks $\mathcal{S}$ and $\mathcal{S}'$ of modules over algebras of formal deformation $\mathcal{A}$ on $X$ and $\mathcal{A}'$ on $Y$, respectively. Suppose also we are given a functor $Φ$ from the category of open subsets of $X$ to the category of open subsets of $Y$ together with a functor $F$ of prestacks from $\mathcal{S}$ to $\mathcal{S}'\circΦ$. Then we give conditions for the existence of a canonical functor, extension of $F$ to the category of coherent ${\sha}$-modules such that the cohomology associated to the action of the formal parameter $\hbar$ takes values in $\mathcal{S}$. We give an explicit construction and prove that when the initial functor $F$ is exact on each open subset, so is its extension. Our construction permits to extend the functors of inverse image, Fourier transform, specialization and microlocalization, nearby and vanishing cycles in the framework of $\shd[[\hbar]]$-modules. We also obtain a Cauchy-Kowalewskaia-Kashiwara theorem in the non-characteristic case as well as comparison theorems for regular holonomic $\shd[[\hbar]]$-modules and a coherency criterion for proper direct images of good $\shd[[\hbar]]$-modules.

math.AG↗
arXiv2006

Functorial properties of the microsupport and regularity for ind-sheaves

The notion of microsupport and regularity for ind-sheaves was introduced by M. Kashiwara and P. Schapira in "Microlocal study of ind-sheaves I: microsupport and regularity". In this paper we study the behaviour of the microsupport under several functorial operations and characterize "microlocally" the ind-sheaves that are regular along involutive manifolds. As an application we prove that if a coherent D-module M is regular along an involutive manifold V (in the sense of "Systems of differential equations with regular singularities and their boundary value problems") then the ind-sheaf of temperate holomorphic solutions of M is regular along V. Another application is the notion of microsupport for sheaves on the subanalytic site, which can be given directly, without using the more complicated language of ind-sheaves.

math.AG↗
arXiv2006

Temperate holomorphic solutions and regularity of holonomic D-modules on curves

Let $X$ be a complex manifold. In "Microlocal study of Ind-sheaves I: microsupport and regularity", M. Kashiwara e P. Schapira made the conjecture that a holonomic D-module $\shm$ is regular holonomic if and only if $R\mathcal{I}{hom}_{β_X\shd_X}(β_X\shm,\sho_X^t)$ is regular (in the sense of "Microlocal study of Ind-sheaves I: microsupport and regularity"), the "only if" part of this conjecture following immediately from "Microlocal study of Ind-sheaves I: microsupport and regularity". Our aim is to prove this conjecture in dimension one.

math.AG↗
arXiv2005

Truncated microsupport and hyperbolic inequalities

We prove that the k-truncated microsupport of the specialization of a complex of sheaves $F$ along a submanifold is contained in the normal cone to the conormal bundle along the k-truncated microsupport of $F$. In the complex case, applying our estimates to $F=\text{R}\cal{H}\text{om}_{\cal{D}}(\cal{M}, \sho)$, where $\cal{M}$ is a coherent $\cal{D}$-module, we obtain new estimates for the truncated microsupport of real analytic and hyperfunction solutions. When $\cal{M}$ is regular along $Y$ we also obtain estimates for the truncated microsupport of the holomorphic solutions of the induced system along $Y$ as well as for the nearby-cycle sheaf of $\shm$ when $Y$ is a hypersurface.

math.AG↗