Length of $D_Xf^{-\alpha}$ in the isolated singularity case

TL;DR

This paper derives a formula for the length of a specific ${ m D}_X$-module associated with an isolated singularity of a convergent power series, generalizing previous results and providing conditions related to semi-weighted-homogeneous cases.

Contribution

It introduces a new explicit formula for the length of ${ m D}_Xf^{-eta}$ in the isolated singularity case, extending prior work to more general settings.

Findings

Provides a formula involving $\widetilde{\nu}_{\alpha}$, $r_f$, and $\widetilde{\delta}_{\alpha}$

Generalizes previous results by Bitoun and Schedler to broader cases

Offers conditions for conjectures in semi-weighted-homogeneous scenarios

Abstract

Let $f$ be a convergent power series of $n$ variables having an isolated singularity at 0. For a rational number $\alpha$, setting $(X,0)=({\mathbb C}^n,0)$, we show that the length of the ${\mathcal D}_X$-module ${\mathcal D}_Xf^{-\alpha}$ is given by $\widetilde{\nu}_{\alpha}+r_f\widetilde{\delta}_{\alpha}+1$. Here $r_f$ is the number of local irreducible components of $f^{-1}(0)$ (with $r_f=1$ for $n>2$), $\widetilde{\nu}_{\alpha}$ is the dimension of the graded piece ${\rm Gr}_V^{\alpha}$ of the $V$-filtration on the saturation of the Brieskorn lattice modulo the image of $N:=\partial_tt-\alpha$ on ${\rm Gr}_V^{\alpha}$ of the Gauss-Manin system, and $\widetilde{\delta}_{\alpha}:=1$ if $\alpha\in{\mathbb Z}_{>0}$, and 0 otherwise. This theorem can be proved also by employing a generalization a recent formula of T. Bitoun in the integral exponent case. The theorem generalizes an assertion by T. Bitoun and T. Schedler in the weighted homogeneous case where the saturation coincides with the Brieskorn lattice and $N=0$. In the semi-weighted-homogeneous case, our theorem implies some sufficient conditions for their conjecture about the length of ${\mathcal D}_Xf^{-1}$ to hold or to fail.