Test modules, singularities and nearby cycles

• Applicant: Privatdozent Dr. Axel Stäbler
• Subject Area: Mathematics
• Term: from 2016 to 2017
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 316646288

Over the complex numbers there are intimate connections between the theory of multiplier ideals, (holonomic) D-modules and the nearby cycles functor. There is evidence that such a connection should also exist in positive characteristic. The analogues of these objects in positive characteristic are test modules, unit F-modules and the p-adic (or mod p) nearby cycles functor. My project is concerned with an investigation of this connection.In characteristic zero the link between D-modules and the nearby cycles functor is provided by the so-called V-filtration. In previous work I established an analogue of this V-filtration in characteristic p under a technical condition, called F-regularity. In fact, in positive characteristic the construction is carried out in the category of Cartier modules (where test modules "live"). This category has no direct analogue over the complex numbers. Part of my project is to remove the above mentioned technical restriction on the construction of the V-filtration and to also investigate whether -- in analogy with the complex numbers -- we can define this filtration directly in the category of unit F-modules. Over the complex numbers the concept of intermediate extensions is crucial for the theory of holonomic D-modules since any holonomic D-module can be built up from simple components via intermediate extensions. In characteristic p intermediate extensions are directly related to the theory of test modules. This suggests a way to approach functoriality questions of intermediate extensions that are known in characteristic zero but open in characteristic p.A shortcoming of the present state of the theory of test modules is that their construction is not functorial. In fact, the technical condition F-regularity mentioned above does guarantee functoriality. We intend to modify the definition of the test module in order to achieve functoriality in full generality. We expect that most results for the existing theory of test modules can be carried over to such a modified version. Finally, in characteristic zero the so-called jumping numbers of the multiplier ideal filtration yield information on the eigenvalues of the monodromy action on the Milnor fiber of the singularity. There is evidence that there should exist a characteristic p (or rather p-adic) analogue of this phenomenon. We intend to investigate this in the explicit case of ADE-singularities and normal crossing divisors using Monsky-Washnitzer cohomology which is very accessible to explicit computations.

• DFG Programme: Research Fellowships
• International Connection: USA
• Host: Professor Mircea Mustata, Ph.D.