Hodge Filtration on Local Cohomology and Minimal Exponents

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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.