l-进层的邻近圈的分歧界

The far-reaching l-adic cohomology theory designed by Grothendieck is a very important tool in the study of the arithmetic and the geometry of positive characteristic varieties and varieties over p-adic fields. The computation of cohomologies of l-adic sheaves can be often translated to the study of its nearby cycles complexes. Most former crucial researches on nearby cycles focus on the alternating sum of the rank (or the Swan conductor) of every cohomology of its stalk. The goal of this project is to study each cohomology of nearby cycles complexes. In particular, we will mainly consider the ramification bound in terms of the Galois representation associated to nearby cycles complexes. In one of my former work with collaborators, we gave such a bound for l-adic sheaves with wild ramifications on a variety over an equal characteristic local field with good reduction, in help of the characteristic cycles theory. In this project, we would like to continue our study and we try to give a ramification bound of the nearby cycles of l-adic sheaves on varieties over p-adic field with semi-stable reduction. It will be more interested and more important in arithmetic. Our key idea is to generalize the characteristic cycles theory for l-adic sheaves on smooth positive characteristic varieties and its many properties to varieties over p-adic fields under good geometric conditions, and to study our project using the generalized theory.

Grothendieck提出的影响深远的l-进上同调理论是研究正特征簇以及p-进域簇上算术与几何的重要工具。计算l-进层的上同调往往可以转化为研究层的邻近圈复形。前人对l-进层邻近圈的重要研究结果侧重于考虑其芽的每一阶上同调的秩（或各阶上同调Swan导子）的交错和。本项目的研究目标是研究邻近圈的各阶上同调。特别的，我们主要考虑l-进层的邻近圈在各阶上同调伽罗华作用下的分歧界。在本人与合作者先前的工作中，在等特征具有良好约化的代数簇上具有野分歧的l-进层，我们利用l-进层的特征圈理论研究了这一分歧结构，得到了一个良好的分歧界。在本项目中，我们希望给出有半稳定约化的p-进簇上的任意l-进层的邻近圈的分歧界。这在算术上是更加令人关心的重要问题。我们的关键想法是在好的几何条件下，将光滑正特征簇上l-进层的特征圈理论及其诸多性质推广到p-进簇，并利用这一推广的理论来研究本项目的课题。

l-进上同调理论是数论与代数几何研究中的重要工具。研究l-进层的邻近圈复形的结构在l-进上同调的计算中起到了关键作用。I. Leal在2016年对局部域基底代数簇上l-进层提出了上同调分歧界猜想。我们对l-进层的邻近圈也提出了类似的分歧界猜想。上述两个猜想在未来研究l-进层的贝蒂数的界中将起到重要作用，并能应用于指数和的估计中。围绕上述猜想我们在本项目中取得了一系列进展。首先，我们在具有稳定约化的几何情形证明了上述猜想，并以此给出了曲线基底的Abel概型上平展层的导子公式。随后在一篇预印本论文中，我们在几何情形完全证明了上述猜想。我们所用的主要工具是A. Beilinson和T. Saito在2016年前后分别发展出来的平展层的奇异支撑和特征圈理论。D-模理论与l-进层理论有某种程度的相似性。利用J.-P. Teyssier发展的上述分歧界结果在D-模理论中的类比，我们在一篇预印本论文中证明了光滑复曲面上给定非正则数与秩的亚纯平坦联络贝蒂数的有界性。