New slope inequalities for families of complete intersections

We prove f -positivity of OX (1) for arbitrary dimensional fibrations over curves f : X → B whose general fibre is a complete intersection. In the special case where the family is a global complete intersection, we prove numerical sufficient and necessary conditions for f -positivity of powers of OX (1) and for the relative canonical sheaf. From these results we also derive a Chow instability condition for the fibres of relative complete intersections in the projective bundle of a μ−unstable bundle.