Higher Du Bois and higher rational singularities

Higher Du Bois and higher rational singularities

This workshop, sponsored by AIM and the NSF, will be devoted to studying recently developed, natural generalizations of the classical notions of Du Bois and rational singularities of complex algebraic varieties. Classes of mild singularities often satisfy certain vanishing properties which are known for smooth varieties. These vanishing results are applied in Moduli theory, Hodge theory and heavily in the MMP. Recently "higher" versions of the cohomologically defined singularities emerged from various points of view: Mustaţă and Popa were interested in deeper adjunction properties and local vanishing results; Laza and Friedman studied deformations of Calabi-Yau and Fano varieties; and finally in the study by Kovács, Kollár and Taji of Hodge bundles. The aim of this workshop is to bring together foremost experts in the field with the goal of settling foundational issues and to establishing the scope and applicability of these new ideas.

At this point, we have a fairly good understanding of rational and Du Bois singularities (in char = 0). The higher rational and higher Du Bois singularities are straightforward extensions of the classical notions, however many foundational issues remain in the theory outside of the local complete intersection (LCI) case. One major goal of the workshop is to investigate the subtleties of the non-LCI case and provide a satisfying theory in this more general situation. This would have immediate applications to the study of singular hyperKähler varieties and also to the Hodge theory of local cohomology modules. In another direction of generalization, it is natural to ask for analogous classes of singularities in positive and mixed characteristic. Even in the classical case, the corresponding notions of Du Bois and rational singularities are not so obvious to define. One of the main difficulties is the lack of Hodge theory in these situations. There has been a ton of interesting work in this direction in recent years, using the Frobenius morphism in positive characteristic and using perfectoid geometry in mixed characteristic.

A final goal is toward applications of the theory. In the classical settings, Du Bois singularities are the largest class of mild singularities and the most technical, however they turn out to be the most relevant to applications to moduli problems and Hodge theory. In the study of moduli spaces of higher dimensional varieties (especially in the Fano and Calabi-Yau case) the higher Du Bois and higher rational singularities naturally show up. These applications are only the beginning of the story. We expect that many more applications will emerge in the near future. Two directions that we plan to explore as part of the workshop are the study of the behavior of degenerations of Hodge structures with mild singularity hypotheses on the fibers and the behavior of Hodge bundles.

In summary, the main topics for the workshop are

1. Investigating higher Du Bois and higher rational singularities outside the LCI setting.
2. Developing a definition of higher Du Bois and higher rational singularities in the posi- tive/mixed characteristic settings.
3. Applying the theory to moduli and Hodge theory. Finding new applications.

This event will be run as an AIM-style workshop. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

For more information email workshops@aimath.org

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