Relative Severi inequality for fibrations of maximal Albanese dimension over curves

Let $f: X \to B$ be a relatively minimal fibration of maximal Albanese dimension from a variety $X$ of dimension $n \ge 2$ to a curve $B$ defined over an algebraically closed field of characteristic zero. We prove that $K_{X/B}^n \ge 2n! \chi_f$, which was conjectured by Barja in [2]. Via the strategy outlined in [5], it also leads to a new proof of the Severi inequality for varieties of maximal Albanese dimension. Moreover, when the equality holds and $\chi_f > 0$, we prove that the general fiber $F$ of $f$ has to satisfy the Severi equality that $K_F^{n-1} = 2(n-1)! \chi(F, \omega_F)$. We also prove some sharper results of the same type under extra assumptions.Comment: Comments are welcom