Microsupport of tempered solutions of D-Modules associated to smooth morphisms

Abstract

Let $f:X\to Y$ be a smooth morphism of complex analytic manifolds and let $F$ be an $\mathbb{R}$-constructible complex on $Y$. Let $\mathcal{M}$ be a coherent $\shd_X$-module. We prove that the microsupport of the solution complex of $\shm$ in the tempered holomorphic functions $t \shh \text{om} (f^{-1} F, \sho_X)$, is contained in the 1-characteristic variety of $\mathcal{M}$ associated to $f$, and that the microsupport of the solution complex in the tempered microfunctions $tμhom(f^{-1}F, \sho_X)$ is contained in the 1-microcharacteristic variety of the microlocalized of $\shm$ along $T^{\ast }Y{\times }_{Y}X$. This applies in particular to the complex of solutions of $\shm$ in the sheaf of distributions holomorphic in the fibers of an arbitrary smooth morphism.

Final version to appear in Houston Journal of Mathematics with a dedicatory added

References in corpus (1)

• Micro-support and Cauchy problem for temperate solutions of regular $\mathcal{D}$-Modules