A Flatness Property for Filtered $\mathcal D$-modules

Abstract

Let M be a coherent module over the ring D X of linear differential operators on an analytic manifold X and let Z 1 , . . . , Z k be k germs of transverse hypersurfaces at a point x ∈ X. The Malgrange-Kashiwara V-filtrations along these hypersurfaces, associated with a given presentation of the germ of M at x, give rise to a multifiltration U • (M) of M x as in Sabbah's paper [9] and to an analytic standard fan in a way similar to [3]. We prove here that this standard fan is adapted to the multifiltration, in the sense of C. Sabbah. This result completes the proof of the existence of an adapted fan in [9], for which the use of

Bernstein–Sato polynomials of arbitrary varieties

“…1.1, because our definition requires certain additional binomial polynomials as in (2.10) below, and Theorem 2 does not hold for that polynomial without the additional term, see (5.4) below. In [27] Sabbah proved the existence of nonzero polynomials of several variables which satisfy a functional equation similar to the above one, see also [1], [8], [13]. However, its relation with b f (s) seems to be quite nontrivial, see (5.1) below.…”