Stability conditions and positivity of invariants of fibrations

Abstract

We study three methods that prove the positivity of a natural numerical invariant associated to $1-$parameter families of polarized varieties. All these methods involve different stability conditions. In dimension 2 we prove that there is a natural connection between them, related to a yet another stability condition, the linear stability. Finally we make some speculations and prove new results in higher dimension.Comment: Final version, to appear in the Springer volume dedicated to Klaus Hulek on the occasion of his 60-th birthda