Tatsuro Kawakami Tatsuro Kawakami

Associate Professor, Graduate School of Mathematical Sciences, The University of Tokyo *Profile is at the time of the award.

2026HagukumuScience & Engineering(hagukumu)

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Research topics: Study of the Extension Problem for Differential Forms in Positive Characteristic

Keyword

Summary: The theme of this research is the extension problem for differential forms in positive characteristic. The extension problem asks whether differential forms can be lifted locally to a birational model such as a resolution of singularities, and it is a fundamental problem related to the properties of differential forms and singularities of algebraic varieties._x000D_
The goal of this research is to clarify whether the important class of singularities in positive characteristic known as F-singularities satisfies the extendability of higher differential forms.

Message

In this project, we study the extension problem for differential forms on algebraic varieties in positive characteristic. The extension problem for differential forms asks to what extent differential forms on a singular algebraic variety can be lifted to a birational model such as a resolution of singularities. This problem is a fundamental theme connecting the study of singularities with birational geometry._x000D_
In the case of complex algebraic varieties, after a long history of research, very powerful results on this problem have been obtained in recent years. These results are mainly proved using Hodge theory as a central tool. However, Hodge theory is a theory built upon the properties of the complex numbers, and at present there is no corresponding theory available in the setting of positive characteristic. This is one of the reasons why many problems in algebraic geometry in positive characteristic remain less understood compared to the complex case._x000D_
In this project, we aim to analyze the extension problem for differential forms in positive characteristic using techniques specific to positive characteristic, such as the Cartier operator. If phenomena that are understood over the complex numbers via Hodge theory can be explained using methods intrinsic to positive characteristic, we expect that this approach will provide new perspectives for the development of algebraic geometry in positive characteristic.