Algebraic slopes and analytic slopes of a \({\mathcal D}\)-module. (Pentes algébriques et pentes analytiques d’un \({\mathcal D}\)-module.) (French) Zbl 0944.14007

Laurent, Yves; Mebkhout, Zoghman

Ann. Sci. Éc. Norm. Supér. (4) 32, No. 1, 39-69 (1999).

Let \(Y\) be a (smooth) hypersurface in a complex analytic variety \(X\) over \({\mathbb C}\) (or an algebraically closed field of characteristic zero). To a holonomic \({\mathcal D}\) - module \(M\) one can associate an irregular perverse sheaf \(\text{Irr}_Y(M)\) with a filtration indexed by \(s \in [1,\infty[\) such that the associated graded sheaves \(\text{Gr}(s)\) form a locally finite family. This defines (at each \(y\in Y\)) the transcendental slopes of \(M\). This filtration also enables one to define the Newton polygon of \(M\). On the other hand, the theory of microcharacteristic varieties [Y. Laurent, Ann. Sci. Éc. Norm. Supér., IV. Sér. 20, 391-441 (1987; Zbl 0646.58021)] associates to \(M\) the microcharacteristic cycles in the cotangent bundle \(T^* N\) of the normal bundle \(N\) of \(Y\) in \(X\). Projecting them to \(Y\), one defines the algebraic slopes of \(M\). The main results of the paper are the following:
(I) Comparison theorem. \( s\in ]1,\infty[\) is a transcendental slope of \(M\) at \(y\in Y\) if and only if it is an algebraic slope of \(M\) at \(y\).
(II) Integrality theorem. The function \(s \chi (\text{Gr}(s))\) has integral values. The Newton polygon of \(M\) has integral vertices.

Reviewer: U.N.Bhosle (Bombay)

MSC:

14F10	Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14M25	Toric varieties, Newton polyhedra, Okounkov bodies

Keywords:

D-module; analytic slope; microcharacteristic varieties; transcendental slopes; Newton polygon

Citations:

Zbl 0646.58021

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