The constant scalar curvature K\"ahler condition is very general

Abstract

Recent work of Trusiani implies that the existence of a constant scalar curvature K\"ahler metric on a smooth polarised variety with discrete automorphism group is equivalent to uniform arc K-stability. We prove that uniform arc K-stability is essentially algebraic in flat families of polarised varieties. When the polarised varieties are further smooth and have discrete automorphism group, combining these two results implies that the constant scalar curvature K\"ahler locus is very general. We use this result to give the first examples of constant scalar curvature K\"ahler metrics whose existence only follows from the recent solution of the Yau--Tian--Donaldson conjecture. Our technique is to prove a general result stating that stability of a pair in the sense of Paul is essentially an algebraic property in families, and to employ prior work with Reboulet relating uniform arc K-stability to stability of an associated pair.