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

TL;DR

This paper investigates the microsupport of tempered solutions of D-modules associated with smooth morphisms, establishing containment results related to 1-characteristic and microcharacteristic varieties, with applications to fiberwise holomorphic distributions.

Contribution

It proves new containment results for the microsupport of tempered solutions of D-modules in terms of 1-characteristic and microcharacteristic varieties, extending understanding of solution complexes in complex geometry.

Findings

Microsupport of tempered solutions is contained in the 1-characteristic variety.

Microsupport in tempered microfunctions is contained in the 1-microcharacteristic variety.

Results apply to solutions in sheaves of fiberwise holomorphic distributions.

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 $\cal{M}$ 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 $\cal{M}$ associated to $f$, and that the microsupport of the solution complex in the tempered microfunctions $t\mu hom(f^{-1}F, \sho_X)$ is contained in the 1-microcharacteristic variety of the microlocalized of $\shm$ along $T^*Y\times_Y X$. 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.