Inversion of subadjunction and multiplier ideals

TL;DR

This paper generalizes the inversion of adjunction theorem using multiplier ideals to centers of arbitrary codimension, providing a new perspective through subadjunction and defining an adjoint ideal to measure non-klt behavior.

Contribution

It extends Kawamata's subadjunction theorem to arbitrary codimension centers by introducing an adjoint ideal that captures non-klt properties outside the generic point.

Findings

The adjoint ideal restricts to the multiplier ideal on the center Z.

The boundary in Kawamata's subadjunction is klt iff Z is an exceptional log-canonical center.

The theorem generalizes the restriction theorem to arbitrary codimension centers.

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).