Stability conditions and positivity of invariants of fibrations

TL;DR

This paper explores three stability-based methods to establish positivity of invariants in fibrations, connecting them in dimension 2 and extending results and conjectures to higher dimensions.

Contribution

It unifies three stability approaches in dimension 2 and advances the understanding of invariants and stability conditions in higher-dimensional fibrations.

Findings

Established connections between stability methods in dimension 2.

Proved new results relating to positivity of invariants in higher dimensions.

Made conjectures and provided partial results for higher-dimensional cases.

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.