Hodge-Theoretic Generalizations of Multiplier Ideals

这项研究涉及代数几何的几个项目。代数几何是研究多项式方程组的解的经典学科，它在数学的许多领域中起着关键作用，无论是纯数学还是应用数学。多项式方程组的解空间具有非常丰富的结构，可以是光滑（连续）或奇异（不连续）的。这个项目关注的是一种新的方法来奇点，从复杂的几何转移到代数设置的想法。主要目标是进一步发展“霍奇理想”理论，这是奇异点的微妙不变量。 更详细地说，几个方向的研究将追求：扩展当前版本的霍奇理想与约化超曲面（人们希望扩展到Q-因子或定义余维为1的子模式的理想）;将Hodge滤子的研究推广到一个单元上（等价于Hodge理想的研究）推广到正则局部环沿着任意理想的局部上同调模上的Hodge滤子的研究;研究了Hodge理想的一个潜在的正特征模，探讨了Hodge理想与Schuelet-Schuermann-Yokura的motivic Chern变换之间的联系;并开发基于更经典方法的工具来给出霍奇理想结果的新证明，其当前的证明依赖于齐藤的霍奇理论模块。

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.