Birational geometry of irregular varieties in zero and positive characteristic

In this dissertation, we mainly focus on constructing two results that characterize cer- tain varieties by their birational invariants. In characteristic 0, we generalize a celebrated theorem of Kawamata by showing that for a projective log canonical pair (X, ∆), if the Kodaira dimension of KX + ∆ is 0 and the dimension of the Albanese variety Alb(X) of X is equal to the dimension of X, then X is birational to an abelian variety. In characteristic p > 0, we show a classification result for surfaces of general type beyond the Noether line. More precisely, suppose that S is a minimal projective surface in characteristic p ≥ 11, χ(OS) = 1 and dim(Alb(S)) = 4, and S lifts to the second Witt vectors. Then under mild assumption on the Albanese variety and the Albanese morphism of S, S is a product of two smooth curves of genus 2.