Séminaire de Mathématique
# Bernstein-Sato Theory in Positive Characteristic and Unit Root Nearby Cycles

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Europe/Paris

Amphithéâtre Léon Motchane (IHES)
### Amphithéâtre Léon Motchane

#### IHES

Le Bois Marie
35, route de Chartres
91440 Bures-sur-Yvette

Description

Séminaire de géométrie arithmétique

In the setting of algebraic geometry in characteristic zero (or of complex geometry), the Bernstein-Sato polynomial is a polynomial defined for a function on a smooth variety and has deep connections with several invariants attached to the singularities of the zero locus of the function, among which the focus in the talk is on the connection with the monodromy eigenvalues on the nearby cycle sheaf, known as the theorem of Kashiwara and Malgrange.

There have been attempts to develop a Bernstein-Sato-type theory also in the setting of positive characteristic, and these have led to a definition of Bernstein-Sato roots (but not their multiplicities), which again has deep connections to the theory of singularities. However, it has not been well studied how this theory is related to the monodromy eigenvalues on a nearby cycle sheaf. In this talk, I will explain an observation that the Bernstein-Sato roots seem to recover some of the monodromy eigenvalues on a suitable nearby cycle sheaf but only those on the unit root part, which we think suggests a better definition of Bernstein-Sato roots that captures all the monodromy eigenvalues and produces finer information about the singularities of the zero locus. This is a joint work in progress with Eamon Quinlan-Gallego and Daichi Takeuchi.

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Organized by

Ahmed Abbes

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