A Partial Comparison of Stability Notions in Kähler Geometry

Abstract

In this follow up work to Dyrefelt (J Geom Anal, 2017. https://doi.org/10.1007/s12220-017-9942-9), Dervan and Ross (Math Res Lett 24, 2017), Dervan (Math Ann, 2017. https://doi.org/10.1007/s00208-017-1592-5), and Sjostrom Dyrefelt (Int Math Res Not 2018. https://doi.org/10.1093/imrn/rny094) 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 Sjostrom Dyrefelt (Int Math Res Not 2018. https://doi.org/10.1093/imrn/rny094). 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.