中国科学院数学与系统科学研究院
数学研究所
数学科学全国重点实验室
学术报告
代数几何研讨班
Speaker: 张磊（中国科学技术大学）
Inviter: 刘杰
Language: English
Title: K-trivial varieties in characteristic p
Time&Venue: 2026年09月29日（星期二）10:00-11:30 MCM410
Abstract: By a $K$-trivial variety we mean a projective variety $X$ whose canonical divisor $K_X$ is numerically trivial. These varieties are of special interest in the classification theory of algebraic varieties. The first question is whether abundance holds for $X$, equivalently, whether $K_X$ is $\mathbb{Q}$-linearly equivalent to zero. For this purpose, it is natural to apply the Albanese map $a_X : X \to A$. When $X$ is defined over the complex numbers, Kawamata proved that the Albanese morphism $X \to A$ of a $K$-trivial variety $X$ induces a fiber bundle structure over an abelian variety; more precisely, there exists an isogeny $A' \to A$ of abelian varieties such that the base change of $a_X$ along this isogeny splits. This result was generalized by Cao to varieties with $-K_X$ nef, in which positivity plays a central role. In this talk, we shall discuss $K$-trivial varieties in characteristic $p$. A main difference in characteristic $p$ is that the geometric generic fiber of the Albanese morphism can be quite singular. We shall first propose a precise splitting structure, which we expect to hold for $X$ and which has been verified by Patakfalvi, Zdanowicz, and Ejiri under the condition that the geometric generic Albanese fiber is $F$-regular. We attempt to adapt Cao's strategy to characteristic $p$. We develop an induction process that requires working over imperfect fields and developing positivity results on singular varieties.
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