# Yuan Wang (Utah): On the characterization of abelian varieties for log pairs in zero and positive characteristic

Yuan Wang (Utah), gives the Seminar in Algebra and Algebraic Geometry:

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

Let X be a projective variety and D an effective Q-divisor on X. A celebrated theorem of Kawamata says that if X is smooth and k(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 X is a fiber space map. In this talk I will further discuss the relationship between k(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.