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 $\mathrm{lct}(f,J_f^2)>1$ if and only if the hypersurface defined by $f$ has rational singularities. Moreover, if it does not have rational singularities, then $\mathrm{lct}(f,J_f^2)=\mathrm{lct}(f)$. We give two proofs, one relying on arc spaces and one that goes through the inequality $\tilde{\alpha }(f)\ge \mathrm{lct}(f,J_f^2)$, where $\tilde{\alpha }(f)$ 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 $\tilde{\alpha }(f)$ replaced by the motivic oscillation index $\mathrm{moi}(f)$. We also show a part of Igusa's strong monodromy conjecture, for poles larger than $-\mathrm{lct}(f,J_f^2)$. We end with a discussion of lct-maximal ideals: these are ideals $I$ with the property that $\mathrm{lct}(I)<\mathrm{lct}(J)$ for every $J$ with $I⊊J$.

This supersedes arXiv:1901.08111. V.2: final version, to appear in the special volume of Alg. Geom. Phys. in honor of Yuri I. Manin