On the behavior of adjoint ideals under pure morphisms
Shunsuke Takagi, Tatsuki Yamaguchi
TL;DR
This work extends the descent theory of singularities to non-$\mathbb{Q}$-Gorenstein contexts by studying adjoint ideals under pure morphisms. It develops an ultraproduct- and BCM-based framework to characterize adjoint ideals and to handle cycle-theoretic pullbacks, establishing that plt-type singularities descend along pure morphisms under suitable hypotheses and yielding a positive answer to Zhuang's question in the klt case. The key contributions include a characterization of adjoint ideals via ultraproduct BCM objects in the $\mathbb{Q}$-Cartier setting, a generalization of plus closure to divisors, and faithful flat descent results for adjoint ideals, together with counterexamples illustrating the necessity of the hypotheses. These results broaden the scope of MMP-type singularity descent and provide new tools for analyzing non-$\mathbb{Q}$-Gorenstein pairs in complex geometry and birational geometry.
Abstract
We characterize adjoint ideal sheaves via ultraproducts and, utilizing this characterization, study their behavior under pure morphisms. 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 $Γ$ on $X$ without common components, we have the following result: if the cycle-theoretic pullback $E:=f^{\natural}D$ is reduced and $(Y, E+f^*Γ)$ is of plt type along $E$, then $(X, D+Γ)$ is of plt type along $D$. This provides an affirmative answer to a question posed by Z. Zhuang.