Hodge-Theoretic Generalizations of Multiplier Ideals

乘数理想的霍奇理论推广

• 批准号: 1701622
• 负责人: Mircea Mustata
• 金额: $ 36万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1701622&HistoricalAwards=false
• 关键词: Hodge; Theoretic; Generalizations; Multiplier; Ideals

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.

这项研究涉及代数几何中的几个项目。传统上，代数几何是对多项式方程组解的研究，它在数学和应用数学的众多领域中发挥着关键作用。多项式方程组的解空间具有非常丰富的结构，可以是平滑（连续）或奇异（不连续）的。该项目涉及一种解决奇点的新方法，该方法是将思想从复杂几何转移到代数环境中。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，将追求几个研究方向：扩展与简化超曲面相关的霍奇理想的当前版本（人们希望扩展 Q-除数或定义在余维一上简化的子方案的理想）；将单元局部化的霍奇过滤研究（相当于霍奇理想的研究）扩展到正则局部环沿任意理想的局部上同调模的霍奇过滤研究；研究积极特征中霍奇理想的潜在类似物；利用霍奇理想与布拉瑟莱特-舒尔曼-约库拉的动机陈省身变换之间的联系；开发基于更经典方法的工具，为 Hodge 理想提供新的结果证明，其当前的证明依赖于 Saito 的 Hodge 模理论。