# A Partial Comparison of Stability Notions in K\"ahler Geometry

## TL;DR

This paper introduces a restricted geodesic stability concept in Kähler geometry, compares various stability notions, and establishes their equivalences under certain conditions, leading to new examples of K-polystable manifolds.

## Contribution

It develops a framework for restricted geodesic stability, compares multiple stability notions, and proves K-polystability for new classes of manifolds with irrational polarization.

## Key findings

- Equivalence of stability notions under 'weak cscK' condition.
- Partial comparison of algebraic, transcendental, and geodesic stability.
- Identification of new K-polystable manifolds with irrational polarization.

## Abstract

In this follow up work to [45, 33, 32, 46] we introduce and study a notion of geodesic stability restricted to rays with prescribed singularity types. A number of notions of interest fit into this framework, in particular algebraic- and transcendental K-polystability, equivariant K-polystability, and the geodesic K-polystability notion introduced by the author in [46]. We provide a partial comparison of the above notions, and show equivalence of some of these notions provided that the underlying manifold satisfies a certain 'weak cscK' condition. As an application this proves K-polystability of a new family of cscK manifolds with irrational polarization.