A Boundedness theorem for nearby slopes of holonomic D-modules

TL;DR

This paper introduces a new notion of slopes for complex holonomic D-modules using twisted nearby cycles, establishes a boundedness theorem, and applies these results to irregular connections and wild ramification.

Contribution

It defines a novel slope concept for D-modules, proves a boundedness theorem, and extends the regularity characterization to irregular connections and arithmetic wild ramification.

Findings

Established a boundedness result for the new slopes.

Characterized regularity of D-modules via slopes.

Provided explicit bounds for slopes in families of irregular connections.

Abstract

Using twisted nearby cycles, we define a new notion of slopes for complex holonomic D-modules. We prove a boundedness result for these slopes, study their functoriality and use them to characterize regularity. For a family of (possibly irregular) algebraic connections E\_t parametrized by a smooth curve, we deduce under natural conditions an explicit bound for the usual slopes of the differential equation satisfied by the family of irregular periods of the E\_t. This generalizes the regularity of the Gauss-Manin connection proved by Katz and Deligne. Finally, we address some questions about analogues of the above results for wild ramification in the arithmetic context.