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.