论文标题
原木规范阈值和理性概念
The log canonical threshold and rational singularities
论文作者
论文摘要
我们表明，如果$ f $在光滑的复杂品种$ x $上是一个非零的，不可辨认的功能，而$ j_f $是$ f $的jacobian理想，那么$ {\ rm lct}（f，j_f^2）> 1 $，并且只有当$ f $定义的hypersurface被$ f $定义的时，此外，如果它没有合理的概念，则$ {\ rm lct}（f，j_f^2）= {\ rm lct}（f）$。我们给出了两个证据，一个依靠弧空，一个通过不等式$ \widetildeα（f）\ geq {\ rm lct}（f，j_f^2）$，其中$ \wideTildeα（f）$是$ f $的最低指数。对于$ \ overline {\ mathbf {q}} $的多项式，我们还证明了后者不平等的类似物，$ \widetildeα（f）$替换为动机振荡索引$ {\ rm moi}（f）$。我们还展示了Igusa强烈的单一构想的一部分，用于大于$ - {\ rm lct}（f，j_f^2）$的电线杆。我们以讨论LCT最大理想的讨论结尾：这些是$ {\ rm lct}（i）<{\ rm lct}（j）$的属性的理想$ i $，每个$ j $带有$ i \ i \ subsetneq j $。
We show that if $f$ is a nonzero, noninvertible function on a smooth complex variety $X$ and $J_f$ is the Jacobian ideal of $f$, then ${\rm lct}(f,J_f^2)>1$ if and only if the hypersurface defined by $f$ has rational singularities. Moreover, if it does not have rational singularities, then ${\rm lct}(f,J_f^2)={\rm lct}(f)$. We give two proofs, one relying on arc spaces and one that goes through the inequality $\widetildeα(f)\geq{\rm lct}(f,J_f^2)$, where $\widetildeα(f)$ is the minimal exponent of $f$. In the case of a polynomial over $\overline{\mathbf{Q}}$, we also prove an analogue of this latter inequality, with $\widetildeα(f)$ replaced by the motivic oscillation index ${\rm moi}(f)$. We also show a part of Igusa's strong monodromy conjecture, for poles larger than $-{\rm lct}(f,J_f^2)$. We end with a discussion of lct-maximal ideals: these are ideals $I$ with the property that ${\rm lct}(I)<{\rm lct}(J)$ for every $J$ with $I\subsetneq J$.
