Inversion of subadjunction and multiplier ideals

Abstract

We present a generalization of the multiplier ideal version of inversion of adjunction, often known as the restriction theorem, to centers of arbitrary codimension. We approach inversion of adjunction from the subadjunction point of view. Let X be a smooth complex projective variety and let Z be an exceptional log-canonical center of an effective Q-divisor D on some dense open subset of X that contains the generic point of Z. Any subvariety of X can be expressed as such a center for some D. We define an adjoint ideal that measures how non-klt (X, D) is outside the generic point of Z. Our main theorem is that this adjoint ideal restricts on Z to the multiplier ideal of an appropriate boundary constructed in the same manner as the boundary in Kawamata's subadjunction theorem. Our theorem extends Kawamata's subadjunction theorem and implies that, in general, the boundary in Kawamata's subadjunction is klt if and only if Z is an exceptional log-canonical center of (X, D).