Arc K-semistability is a very general property
Ruadhaí Dervan
TL;DR
The paper proves that arc K-semistability is a very general property in flat families of polarised varieties, with a parallel result for uniform arc K-stability. It does so by establishing that semistability (and stability) of pairs, in the sense of Paul, are Zariski-open (and very general) properties, then relating arc K-stability to semistability of an associated pair via a Hilbert-scheme construction and the CM line bundle. This provides a broad, algebraic framework to deduce stability properties in families and enables the construction of smooth uniformly arc K-stable varieties that are not known to admit constant scalar curvature Kähler metrics. The approach connects arc stability to stability of pairs and exploits Hilbert scheme techniques to handle singular and general polarised varieties, extending the scope of K-stability-inspired moduli questions beyond Fano varieties.
Abstract
We prove that arc K-semistability is a very general property in flat families of polarised varieties, and prove a similar result for uniform arc K-stability. This can be used to produce the only current examples of smooth uniformly arc K-stable varieties which are not known to admit a constant scalar curvature Kähler metric. Our technique is to prove a general result stating that semistability of a pair in the sense of Paul is a Zariski open property, and to employ prior work with Reboulet relating arc K-semistability to semistability of an associated pair.