论文标题
关于Bitoun和Schedler的猜想
On a conjecture of Bitoun and Schedler
论文作者
论文摘要
假设$ x $是尺寸$ \ geq 3 $和$ f $的平滑复杂的代数种类，定义了$ x $中的hypersurface $ z $，带有独特的单点$ p $。 Bitoun和Schedler猜想$ {\ Mathcal d} $ - 由$ \ tfrac {1} {f} $生成的模块的长度等于$ g_p（z）+2 $，其中$ g_ {p}（z）$是$ z $ s $ z $ in $ p $。我们证明，此长度始终为$ \ geq g_p（z）+2 $，并且当且仅当$ \ tfrac {1} {1} {f} $时，均位于$ {\ Mathcal d} $ - 由$ i_0（f）\ tfrac {1} $ IS $ IS $ IS $ IS $ IS $ i_0（ j}（f^{1-ε}）$，带有$ 0 <ε\ ll 1 $。特别是，我们看到，如果对$（x，z）$是log canonical，则猜想存在。我们还可以简单地恢复Bitoun和Schedler的结果，说该猜想是加权均匀孤立的奇异点。另一方面，我们给出一个示例（$ 3 $变量的多项式，具有普通的单数多重点$ 4 $）。
Suppose that $X$ is a smooth complex algebraic variety of dimension $\geq 3$ and $f$ defines a hypersurface $Z$ in $X$, with a unique singular point $P$. Bitoun and Schedler conjectured that the ${\mathcal D}$-module generated by $\tfrac{1}{f}$ has length equal to $g_P(Z)+2$, where $g_{P}(Z)$ is the reduced genus of $Z$ at $P$. We prove that this length is always $\geq g_P(Z)+2$ and equality holds if and only if $\tfrac{1}{f}$ lies in the ${\mathcal D}$-module generated by $I_0(f)\tfrac{1}{f}$, where $I_0(f)$ is the multiplier ideal ${\mathcal J}(f^{1-ε})$, with $0<ε\ll 1$. In particular, we see that the conjecture holds if the pair $(X,Z)$ is log canonical. We can also recover, with an easy proof, the result of Bitoun and Schedler saying that the conjecture holds for weighted homogeneous isolated singularities. On the other hand, we give an example (a polynomial in $3$ variables with an ordinary singular point of multiplicity $4$) for which the conjecture does not hold.
