Functorial properties of the microsupport and regularity for ind-sheaves

Truncated microsupport and hyperbolic inequalities

We prove that the k-truncated microsupport of the specialization of a complex of sheaves $F$ along a submanifold is contained in the normal cone to the conormal bundle along the k-truncated microsupport of $F$. In the complex case, applying our estimates to $F=\text{R}\cal{H}\text{om}_{\cal{D}}(\cal{M}, \sho)$, where $\cal{M}$ is a coherent $\cal{D}$-module, we obtain new estimates for the truncated microsupport of real analytic and hyperfunction solutions. When $\cal{M}$ is regular along $Y$ we also obtain estimates for the truncated microsupport of the holomorphic solutions of the induced sy…

Temperate holomorphic solutions and regularity of holonomic D-modules on curves

Let $X$ be a complex manifold. In "Microlocal study of Ind-sheaves I: microsupport and regularity", M. Kashiwara e P. Schapira made the conjecture that a holonomic D-module $\shm$ is regular holonomic if and only if $R\mathcal{I}{hom}_{β_X\shd_X}(β_X\shm,\sho_X^t)$ is regular (in the sense of "Microlocal study of Ind-sheaves I: microsupport and regularity"), the "only if" part of this conjecture following immediately from "Microlocal study of Ind-sheaves I: microsupport and regularity". Our aim is to prove this conjecture in dimension one.

Differential forms in positive characteristic II: cdh-descent via functorial Riemann-Zariski spaces

This paper continues our study of the sheaf associated to Kähler differentials in the cdh-topology and its cousins, in positive characteristic, without assuming resolution of singularities. The picture for the sheaves themselves is now fairly complete. We give a calculation $\mathcal{O}_{cdh}(X) \cong \mathcal{O}(X^{sn})$ in terms of the seminormalisation. We observe that the category of representable cdh-sheaves is equivalent to the category of seminormal varieties. We conclude by proposing some possible connections to Berkovich spaces, and $F$-singularities in the last section. The tools dev…

抽象六函子形式主义：扩展到归纳范畴与投射范畴及函子上同调纯性

Abstract Six-Functor Formalisms: Extension to Ind- and Pro- Categories and Functorial Cohomological Purity

本文将抽象六函子形式主义扩展至Ind-与Pro-范畴，应用于定义ind-pro-代数栈的动机稳定同伦理论，并借助Liu-Zheng的多单纯语言证明上同调纯性的函子性。

交换环上的刚性对偶复形及其函子性质

Rigid Dualizing Complexes over Commutative Rings and their Functorial Properties

本文通过刚性对偶复形处理noetherian环的Grothendieck Duality，证明了正则基环上本质有限型环具有唯一的刚性对偶复形，并利用其函子性质构造了扭曲诱导伪函子，为后续在Deligne-Mumford stacks上建立全局对偶性奠定基础。