Hodge Filtration on Local Cohomology and Minimal Exponents

给定一个具有复系数的多个变量多项式，其解集是一个几何对象，在许多有趣的情况下表现出奇点。这些奇点可以通过不变量来测量，这些不变量可以是数值的，也可以是更多涉及的（例如，它们可以依次在多项式环中形成合适的子集）。在过去的几年中，PI利用Hodge理论和微分算子代数理论的工具参与了某些此类不变量的研究。在目前的项目中，PI 计划在多个方向上扩展这项研究。例如，他计划将他之前与 Popa 的工作从由一个方程定义的几何对象奇点的情况推广到更一般的此类奇点。在另一个方向上，他打算研究奇点的一个有趣的数值不变量——最小指数——及其与其他不变量和奇点观点的联系。 PI 将培训研究领域的研究生。PI 一直在与 Popa 一起研究超曲面以及更普遍的 Q 因数的某些奇点不变量、霍奇理想。这些理想可以在 Saito 的混合 Hodge 模块理论的背景下自然地定义。事实证明，霍奇理想的平凡性是由数值不变量（最小指数）控制的，它与双有理几何中的一个重要不变量（对数正则阈值）密切相关。虽然对孤立奇点的最小指数进行了大量研究，但与霍奇理想相关的方法允许处理一般情况。当前项目有两个主要组成部分。在一个方向上，PI 打算与 Popa 一起，通过利用局部上同调的规范过滤，进一步研究霍奇理想理论在超曲面情况之外的扩展。在这方面有几个有趣的问题，涉及与 V 过滤的多变量版本以及与理想的 Bernstein-Sato 多项式的联系。在另一个方向上，PI 计划研究最小指数的一般性质及其与其他奇点观点的关系（通过除数估值和奇点解析，或与动机 zeta 函数相关）。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.