Relative Severi inequality for fibrations of maximal Albanese dimension over curves

Abstract

AbstractLet$f: X \to B$be a relatively minimal fibration of maximal Albanese dimension from a varietyXof dimension$n \ge 2$to a curveBdefined over an algebraically closed field of characteristic zero. We prove that$K_{X/B}^n \ge 2n! \chi _f$. It verifies a conjectural formulation of Barja in [2]. Via the strategy outlined in [4], 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 fibreFoffhas 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.