La construction d'Abbes et Saito pour les connexions m\'eromorphes: aspect formel en dimension 1

TL;DR

This paper adapts Abbes and Saito's geometric measure of wild ramification from l-adic sheaves to differential modules over formal Laurent series, establishing a formula linking their construction to the Levelt-Turrittin decomposition.

Contribution

It introduces a new adaptation of Abbes and Saito's construction for differential modules over characteristic zero fields, connecting it to micro-characteristic cycles.

Findings

Derived a formula relating Abbes and Saito's construction to Levelt-Turrittin decomposition.

Revealed a version of Laurent's micro-characteristic cycles in the algebraically closed case.

Extended the geometric measure of wild ramification to differential modules.

Abstract

By using a blow-up construction, the nearby-cycle functor and l-adic Fourier transform, Abbes and Saito are able to define a geometric measure of wild ramification of l-adic sheaves on the generic point of any complete discrete valuation ring of equal characteristic p with perfect residue field, where p is different from l. In this paper, we adapt their construction to differential modules over the field of formal Laurent series with coefficients in any characteristic zero field K. For such a module M, we prove a formula relating Abbes and Saito's construction to the differential forms occuring in the Levelt-Turrittin decomposition of M. If K is algebraically closed, one recovers a version of Laurent's micro-characteristic cycles.