Microsupport of tempered solutions of \({\mathcal D}\)-modules associated to smooth morphisms (Q2862650)

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 \(M\) be a coherent \(D_X\)-module. The author proves that the microsupport of the solution complex of \(M\) in the tempered hormorphic functions \(t\mu\Hom(f^{-1} F, O_X)\) is contained in the 1-characteristic variety of \(M\) associated to \(f\), and that the microsupport of the solution complex in the tempered microfunctions \(t\mu\Hom(f^{-1}F, O_X)\) is contained in the 1-microcharacteristic variety of the microlocalization of \(M\) along \(T^*Y\times_Y X\), that is the following:NEWLINENEWLINE Theorem. Let \(X\) and \(Y\) be complex manifolds, let \(f: X\to Y\) be a smooth morphism. Then for any \(G\in D^b_{R-c}(C_X\) such that \(SS(G)\subset X\times_YT^*Y\) and any \(M\in\text{Mod}_{\text{coh}}(D_X)\), one has \(SS(R\Hom_{D_X}(M, t\mu\Hom(G, O_X)))\subset\widetilde f^{-1}(C^1_f(M))\).NEWLINENEWLINE This results are applied to complex of solutions of \(M\) in the sheaf of distributions hormorphic in the fibers of an arbitrary smooth morphism.