Hodge-Theoretic Generalizations of Multiplier Ideals

这项研究涉及代数几何中的几个项目。经典地，代数几何是研究多项式方程组解的学科，它在数学的许多领域，无论是纯粹的还是应用的，都起着关键的作用。多项式方程组的解空间具有非常丰富的结构，可以是光滑的（连续的），也可以是奇异的（不连续的）。这个项目关注的是一种新的奇点方法，这种方法是从复杂几何转移到代数设置中产生的。主要目标是进一步发展“霍奇理想”理论，这是奇点的微妙不变量。更详细地说，几个研究方向将被追求：扩展与约简超曲面相关的Hodge理想的当前版本（有人想扩展到q因子或定义在余维为1的子方案的理想）；将单元局部化上Hodge滤波的研究（等价于Hodge理想的研究）推广到沿任意理想的正则局部环的局部上同模上Hodge滤波的研究；研究积极特征中霍奇理想的潜在类比；挖掘霍奇理想与布拉瑟利-舒尔曼-横仓的陈氏转换动机之间的联系；并开发基于更经典方法的工具，对霍奇理想的结果给出新的证明，这些证明目前依赖于齐藤的霍奇模块理论。

This research concerns several projects in algebraic geometry. Classically, algebraic geometry is the study of solutions of systems of polynomial equations, and it plays a key role in numerous fields of mathematics, both pure and applied. The solution space of a system of polynomial equations has a very rich structure that can be smooth (continuous) or singular (discontinuous). This project is concerned with a new approach to singularities that arises from transferring ideas from complex geometry to an algebraic setting. The main goal is to further develop the theory of "Hodge ideals," which are subtle invariants of singularities. In more detail, several directions of research will be pursued: extending the current version of Hodge ideals associated to reduced hypersurfaces (one would like extensions to Q-divisors or to ideals defining a subscheme that is reduced in codimension one); extending the study of Hodge filtrations on localizations at one element (which is equivalent to the study of Hodge ideals) to the study of Hodge filtrations on local cohomology modules of a regular local ring along an arbitrary ideal; investigating a potential analogue of Hodge ideals in positive characteristic; exploiting the connection between Hodge ideals and the motivic Chern transformation of Brasselet-Schuermann-Yokura; and developing tools based on more classical methods to give new proofs of results on Hodge ideals whose current proofs rely on Saito's theory of Hodge modules.