Nearby cycles and composition with a non-degenerate polynomial

Newton 非退化函数的邻近周期公式下降

Descent of nearby cycle formula for Newton non-degenerate functions

本文证明 Newton 非退化函数邻近周期公式及其 motive 版本的下降定理，无需便利条件，并由此推出 Milnor 单值性最大 Jordan 块数公式及单纯形情形下修正的 Steenbrink 猜想。

Composition with a two variable function

We compute the motivic nearby cycles of functions obtained by composition of two functions with distinct sets of variables with a two variable function

The ring of local tropical fans and tropical nearby monodromy eigenvalues

We extend the tropical intersection theory to tropicalizations of germs of analytic sets. In particular, we construct a (not entirely obvious) local version of the ring of tropical fans with a nondegenerate intersection pairing. As an application, we study nearby monodromy eigenvalues -- the eigenvalues of the monodromy operators of singularities, adjacent to a given singularity of a holomorphic function $f$. More precisely, we express some of such values in terms of certain resolutions of $f$. The expression is given it terms of the exceptional divisor strata of arbitrary codimension, general…

The big projective module as a nearby cycles sheaf

We give a new geometric construction of the big projective module in the principal block of the BGG category $\mathscr{O}$, or rather the corresponding $\mathscr{D}$-module on the flag variety. Namely, given a one-parameter family of nondegenerate additive characters of the unipotent radical of a Borel subgroup which degenerate to the trivial character, there is a corresponding one-parameter family of Whittaker sheaves. We show that the unipotent nearby cycles functor applied to this family yields the big projective $\mathscr{D}$-module.

On Milnor fibrations of mixed functions, $a_f$-condition and boundary stability

Convenient mixed functions with strongly non-degenerate Newton boundaries have Milnor fibrations, as the isolatedness of the singularity follows from the convenience. In this paper, we consider the Milnor fibration for non-convenient mixed functions. We also study geometric properties such as Thom's $a_f$ condition, the transversality of the nearby fibers and stable boundary property of the Milnor fibration and their relations.

论Harish-Chandra变换的t-正合性

On a t-exactness property of the Harish-Chandra transform

该研究通过双曲局部化将Vinberg退化附近周环识别为Radon与Harish-Chandra函子的复合，为其正合性提供新简洁证明，推广了相关结果至任意单值群及更一般层论设置。