On the behavior of adjoint ideals under pure morphisms

Abstract

We characterize adjoint ideal sheaves via ultraproducts and study their behavior under pure morphisms by utilizing this characterization. In particular, given a pure morphism $f:Y\to X$ between normal quasi-projective complex varieties, a reduced divisor $D$ and an effective $\mathbb{Q}$-Weil divisor $\Gamma$ on $X$ with no common components, we have the following result: if the cycle-theoretic pullback $E:=f^{♮}D$ is reduced and $(Y,E+f^{\ast }\Gamma )$ is of plt type along $E$, then $(X,D+\Gamma )$ is of plt type along $D$. This provides an affirmative answer to a question posed by Z. Zhuang [Invent. Math. 237 (2024)].