[Submitted on 21 Apr 2025 (v1), last revised 15 Jun 2026 (this version, v3)]

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Abstract:Recent work of Trusiani implies that the existence of a constant scalar curvature Kähler 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 Kähler locus is very general.
We use this result to give the first examples of constant scalar curvature Kähler 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.

Submission history

From: Ruadhaí Dervan [view email]
[v1] Mon, 21 Apr 2025 16:07:21 UTC (15 KB)
[v2] Wed, 3 Jun 2026 16:31:58 UTC (19 KB)
[v3] Mon, 15 Jun 2026 20:12:27 UTC (19 KB)

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