Relative Severi inequality for fibrations of maximal Albanese dimension over curves

摘要

Let f : X -> B be a relatively minimal fibration of maximal Albanese dimension from a variety X of dimension n >= 2 to a curve B defined over an algebraically closed field of characteristic zero. We prove that K-X/B(n) >= 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 fibre 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.

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