Note on algebraic irregular Riemann–Hilbert correspondence

Abstract

The subject of this paper is an algebraic version of the irregular Riemann–Hilbert correspondence which was mentioned in [Tsukuba J. Math. 44 (2020), 155–201]. In particular, we prove an equivalence of categories between the triangulated category {\mathbf{D}}^{\mathrm{b}}_{\mathrm{hol}} (\mathcal{D}_X) of holonomic \mathcal{D} -modules on a smooth algebraic variety X over \mathbb{C} and the triangulated category \mathbf{E}^{\mathrm{b}}_{\operatorname{\mathbb{C}-c}}( \mathrm{I}\mathbb{C}_{X_\infty}) of algebraic \mathbb{C} -constructible enhanced ind-sheaves on a bordered space X^\mathrm{an}_\infty . Moreover, we show that there exists a t-structure on the triangulated category \mathbf{E}^{\mathrm{b}}_{\operatorname{\mathbb{C}-c}}(\mathrm{I}\mathbb{C}_{X_\infty}) whose heart is equivalent to the abelian category of holonomic \mathcal{D} -modules on X . Furthermore, we shall consider simple objects of its heart and minimal extensions of objects of its heart.