Hodge Filtration on Local Cohomology and Minimal Exponents

给定一个多元复系数多项式，它的解集是一个几何对象，在许多有趣的情况下表现出奇异性。这些奇异性可以通过不变量来测量，这些不变量可以是数值的，也可以是更复杂的（例如，它们可以在多项式环中形成合适的子集）。在过去的几年里，PI一直参与研究某些这样的不变量，利用工具从霍奇理论和微分算子的代数理论。在本项目中，PI计划在几个方向上扩展这项研究。例如，他计划将他以前与波帕的工作从由一个方程定义的几何对象的奇点的情况推广到更一般的奇点。在一个不同的方向，他打算调查一个有趣的数值不变量的奇异性-最小指数-及其与其他不变量和观点的连接奇异性。PI将培养研究生的研究领域。PI一直在研究与波帕某些不变量的奇异性，霍奇理想，超曲面，更一般地说，为Q-因子。这些理想可以在Saito的混合Hodge模理论的背景下自然地定义。事实证明，平凡的霍奇理想是由一个数值不变量，最小指数，这是密切相关的一个重要的不变量在双有理几何，对数正则阈值。虽然孤立奇点的最小指数已经被研究了很多，但与霍奇理想相关的方法允许处理一般情况。本项目有两个主要组成部分。在一个方向上，PI打算进一步研究，与波帕，霍奇理想的理论超越超曲面的情况下，通过利用局部上同调的典型过滤的延伸。在这方面有几个有趣的问题，关于与多变量版本的V-过滤和伯恩斯坦-佐藤多项式的理想。在另一个方向，PI计划研究最小指数的一般属性及其与其他奇点观点的关系（通过除法估值和奇点解析，或与动机zeta函数相关）。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估来支持。

Given a polynomial in several variables, with complex coefficients, its solution set is a geometric object that in many interesting situations exhibits singularities. These singularities can be measured by invariants that can be either numerical or more involved (for example, they can form in turn suitable subsets in the polynomial ring). Over the past few years, the PI has been involved in the study of certain such invariants by making use of tools from from Hodge theory and the algebraic theory of differential operators. In the present project, the PI plans to extend this study in several directions. For example, he plans to generalize his previous work with Popa from the case of singularities of geometric objects defined by one equation to more general such singularities. In a different direction, he intends to investigate one interesting numerical invariant of singularities--the minimal exponent--and its connections with other invariants and points of view on singularities. The PI will train graduate students in the area of research.The PI has been studying with Popa certain invariants of singularities, the Hodge ideals, for hypersurfaces and, more generally, for Q-divisors. These ideals can be defined naturally in the context of Saito's theory of mixed Hodge modules. It turns out that the triviality of the Hodge ideals is governed by a numerical invariant,the minimal exponent, which is closely related to an important invariant in birational geometry, the log canonical threshold. While the minimal exponent has been studied a lot for isolated singularities, methods related to Hodge ideals allow treating the general case. There are two main components of the present project. In one direction, the PI intends to further investigate, with Popa, an extension of the theory of Hodge ideals beyond the case of hypersurfaces, by making use of the canonical filtration on local cohomology. There are several interesting questions in this context, concerning connections with the multi-variable version of the V-filtration and with the Bernstein-Sato polynomial for ideals. In a different direction, the PI plans to study general properties of the minimal exponent and its relations to other points of view on singularities (via divisorial valuations and resolution of singularities, or in connection to the motivic zeta function).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.