On canonical bundle formula for fibrations of curves with arithmetic genus one

Abstract In this paper, we develop canonical bundle formulas for fibrations of relative dimension one in characteristic $p>0$ . For such a fibration from a log pair $f\colon (X, \Delta ) \to S$ , if f is separable, we can obtain a formula similar to the one due to Witaszek [36]; if f is inseparable, we treat the case when S is of maximal Albanese dimension. As an application, we prove that for a klt pair $(X,\Delta )$ with $-(K_X+\Delta )$ nef, if the Albanese morphism $a_X\colon X \to A$ is of relative dimension one, then X is a fiber space over A .