A boundedness theorem for nearby slopes of holonomic ${\mathcal{D}}$-modules. (9th September 2016)

• Record Type: Journal Article
• Authors: Teyssier, Jean-Baptiste
• Is Part Of: Compositio mathematica. Volume 152:Number 10(2016)
• Page Start: 2050
• Page End: 2070
• Publication Date: 2016-09-09
• Subjects: 14F10 (primary); D-module, -- nearby cycles, -- rapid decay homology, -- irregular periods; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X16007533
• Languages: English
• ISSNs: 0010-437X

Abstract

Abstract : Using twisted nearby cycles, we define a new notion of slopes for complex holonomic ${\mathcal{D}}$ -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 ${\mathcal{E}}_{t}$ 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 ${\mathcal{E}}_{t}$ . This generalizes the regularity of the Gauss–Manin connection proved by Griffiths, Katz and Deligne.