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

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.