Dimension formulas for the hyperfunction solutions to holonomic \(\mathcal D\)-modules
In the theory of constructible sheaves, as presented by \textit{M. Kashiwara} and \textit{P. Shapira} [cf. their book ``Sheaves on manifolds'' (Grundlehren der Mathematischen Wissenschaften 292, Berlin: Springer Verlag) (1990; Zbl 0709.18001)], a dimension formula is given for the hyperfunction solutions to a certain class of holonomic \({\mathcal D}\)-modules: \[ \dim^{\mathbb{C}}\Hom_{{\mathcal D}_X}({\mathcal M},{\mathcal B}_M)_0= \sum^N_{i=1} \text{mult}_{T^*_{X_i}X}({\mathcal M}) \] for a sheaf \({\mathcal D}_X\) of partial differential operators with holomorphic coefficients on \(X=\) complexification of \(M\), and \({\mathcal M}\) a coherent \({\mathcal D}_X\)-module. The proof is based on index theorems for constructible sheaves. For detailed definitions cf. [\textit{M. Kashiwara}, Systems of microdifferential equations, Progress in Mathematics 34 (Birkhäuser, Boston) (1983; Zbl 0521.58057)].
scientific article; zbMATH DE number 2022085
scientific article