Relative Severi inequality for fibrations of maximal Albanese dimension over curves

Abstract

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.

arXiv commentv2: minor revision, accepted by Forum of Mathematics, Sigma