# On the characterization of abelian varieties for log pairs in zero and positive characteristic

## Yuan Wang (University of Utah)

Let $X$ be a projective variety and $D$ an effective $\mathbb{Q}$-divisor on $X$. A celebrated theorem of Kawamata says that if $X$ is smooth and $\kappa(X)=0$ then the Albanese morphism of $X$ is an algebraic fiber space. Later it was shown by Zhang that if $(X,D)$ is a log canonical pair and $-(K_X+D)$ is nef then the Albanese morphism of any smooth model of $X$ is an algebraic fiber space. In this talk I will further discuss the relationship between $\kappa(K_X+D)=0$, positivity of $-(K_X+D)$ and the Albanese map of $X$ in both characteristic $0$ and characteristic $p>0$, and present some related results. In particular, I will present a result in characteristic $p>0$ and dimension $3$ that is a positive characteristic analog of Zhang's result, and another result in characteristic $0$ that generalizes Kawamata's result to klt pairs.

Tuesday, November 1st, at 4:00pm in HBH 227