The log canonical threshold and rational singularities

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 lct(f, J 2 f ) > 1 if and only if the hypersurface defined by f has rational singularities.Moreover, if it does not have rational singularities, then lct(f, J 2 f ) = lct(f ).We give two proofs, one relying on arc spaces and one that goes through the inequality (f ) lct(f, J 2 f ), where (f ) is the minimal exponent of f .In the case of a polynomial over Q, we also prove an analogue of this latter inequality, with (f ) replaced by the motivic oscillation index moi(f ).We also show a part of Igusa's strong monodromy conjecture, for poles larger than -lct(f, J 2 f ).We end with a discussion of lct-maximal ideals: these are ideals a with the property that lct(a) < lct(b) for every b with a b.