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 ${\mathbf{D}}_{\mathrm{hol}}^{\mathrm{b}}({\mathcal{D}}_X)$ of holonomic $\mathcal{D}$-modules on a smooth algebraic variety $X$ over $\mathbb{C}$ and the triangulated category ${\mathbf{E}}_{\mathbb{C}-c}^{\mathrm{b}}(\mathrm{I}{\mathbb{C}}_{X_{\infty }})$ of algebraic $\mathbb{C}$-constructible enhanced ind-sheaves on a bordered space $X_{\infty }^{\mathrm{an}}$. Moreover, we show that there exists a t-structure on the triangulated category ${\mathbf{E}}_{\mathbb{C}-c}^{\mathrm{b}}(\mathrm{I}{\mathbb{C}}_{X_{\infty }})$ 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.