论文标题
$ d_xf^{ - α} $的长度在孤立的奇异案例中
Length of $D_Xf^{-α}$ in the isolated singularity case
论文作者
论文摘要
令$ f $是$ n $变量的收敛功率系列，其孤立的奇异性在0。对于有理数$α$，设置$（x，0）=（{\ Mathbb c}^n，0）$，我们表明$ {\ Mathcal d} $ _x $ -odule $ iS_______________________________________________________________________________________ $ \widetildeν_α+r_f \widetildeδ_α+1 $。这里$ r_f $是$ f^{ - 1}（0）$的本地不可约组件的数量（$ r_f = 1 $ for $ n> 2 $），$ \widetildeν_α$是分级件$ {\ rm gr} al v $ v $ v $ -filies of the Satrie of the Satrate of the Satrate of the Satrate of the Satrate of的尺寸$ n：= \ partial_tt-α$上的$ {\ rm gr} _v^α$的高斯 - 曼宁系统和$ \widetildeDeΔ_α：= 1 $如果$α\ in {\ mathbb z} _ {> 0} $，and否则0。也可以通过在整体指数案例中采用概括为T. bitoun的最新公式来证明该定理。该定理在加权均匀的情况下通过T. bitoun和T. Schedler概括了一个主张，在该情况下，饱和度与Brieskorn晶格相吻合，$ n = 0 $。在半加权的同质案例中，我们的定理意味着它们的一些足够条件，可以猜测$ {\ mathcal d} _xf^{ - 1} $的长度，以持有或失败。
Let $f$ be a convergent power series of $n$ variables having an isolated singularity at 0. For a rational number $α$, setting $(X,0)=({\mathbb C}^n,0)$, we show that the length of the ${\mathcal D}_X$-module ${\mathcal D}_Xf^{-α}$ is given by $\widetildeν_α+r_f\widetildeδ_α+1$. Here $r_f$ is the number of local irreducible components of $f^{-1}(0)$ (with $r_f=1$ for $n>2$), $\widetildeν_α$ is the dimension of the graded piece ${\rm Gr}_V^α$ of the $V$-filtration on the saturation of the Brieskorn lattice modulo the image of $N:=\partial_tt-α$ on ${\rm Gr}_V^α$ of the Gauss-Manin system, and $\widetildeδ_α:=1$ if $α\in{\mathbb Z}_{>0}$, 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.
