Relative Severi inequality for fibrations of maximal Albanese dimension over curves

Let be a relatively minimal fibration of maximal Albanese dimension from a variety X of dimension to a curve B defined over an algebraically closed field of characteristic zero. We prove that. 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 0$ ]]>, we prove that the general fibre F of f has to satisfy the Severi equality that. We also prove some sharper results of the same type under extra assumptions.