A Boundedness theorem for nearby slopes of holonomic D-modules

Nearby slopes and boundedness for $\ell$-adic sheaves in positive characteristic

The goal of this paper is to motivate a boundedness conjecture on nearby slopes of $\ell$-adic sheaves in positive characteristic, and to prove it for smooth curves. For a constructible $\ell$-adic sheaf, we prove the finiteness of the set of nearby slopes associated to a given morphism. For the constant sheaf, we prove the vanishing of nearby slopes in case of generalized semi-stable reduction.

代数可积连通量的有界不规则性

Algebraic integrable connections with bounded irregularity

本文研究了具有有界不规则性的代数平坦连通量族的有界性，并探讨了全纯D-模族的有界性问题。

斜率有界性与等分布定理

Slope boundedness and Equidistribution theorem

本文证明了特征0函数域上adelic线丛极小斜率的有界性，并据此推导出一般点与小点的等分布定理，同时将方法扩展至数域有限位并证明了$χ$-体积的连续性。

Family of $\mathscr{D}$-modules and representations with a boundedness property

In the representation theory of real reductive Lie groups, many objects have finiteness properties. For example, the lengths of Verma modules and principal series representations are finite, and more precisely, they are bounded. In this paper, we introduce a notion of uniformly bounded families of holonomic $\mathscr{D}$-modules to explain and find such boundedness properties. A uniform bounded family has good properties. For instance, the lengths of modules in the family are bounded and the uniform boundedness is preserved by direct images and inverse images. By the Beilinson--Bernstein corre…

On Irregular Binomial $D$-modules

We prove that a holonomic binomial $D$--module $M_A (I,β)$ is regular if and only if certain associated primes of $I$ determined by the parameter vector $β\in \CC^d$ are homogeneous. We further describe the slopes of $M_A(I,β)$ along a coordinate subspace in terms of the known slopes of some related hypergeometric $D$--modules that also depend on $β$. When the parameter $β$ is generic, we also compute the dimension of the generic stalk of the irregularity of $M_A(I,β)$ along a coordinate hyperplane and provide some remarks about the construction of its Gevrey solutions.