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A boundedness theorem for nearby slopes of holonomic \(\mathcal{D}\)-modules. (English) Zbl 1397.14034

The author discusses the already classical notions of slopes of a holonomic \(\mathcal{D}\)-module along an hypersurface which have been the object of research of many authors in the 70ies-80ies by the crucial role they play in the irregular case. More recently, with the works of Kedlaya and Mochizuki on the notion of good formal structure on flat meromorphic connections, these notions gained a new interest, as well as their functorialities. The author introduces the notion of nearby slope of a \(\mathcal{D}\)-module which he proves is locally bounded. He obtains a new charaterization of the regular holonomic \(\mathcal{D}\)-modules (Theorem 4 and also deduces an interesting estimation of the slopes in the case of a good formal structure along a divisor \(D\). It remains to compare the classical slopes with this new one and also make the relation with the microlocal approach of Yves Laurent.

MSC:

14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
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