The log canonical threshold and rational singularities

Abstract

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 $\widetilde{\alpha}(f)\geq{\rm lct}(f,J_f^2)$, where $\widetilde{\alpha}(f)$ 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 $\widetilde{\alpha}(f)$ 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 $I\subsetneq J$.