A theoretical preprint says a single threshold can organize several outcomes in compact Kähler geometry. Under the paper’s stated assumptions, a positive value is equivalent to the existence of a positive scalar curvature Kähler metric in the class being studied. A value above the class’s average scalar curvature is equivalent to the existence of a unique constant scalar curvature Kähler metric.
The setting is a compact Kähler manifold X with a Kähler class α, under the assumption that the average scalar curvature is positive. The supplied document is an arXiv version 1 preprint dated 20 Aug 2026.
A number built from stability
The paper’s central aim is to show that quantitative data underlying K-stability can also detect whether a fixed Kähler class contains a positive scalar curvature Kähler metric. It works in a transcendental framework using Kähler test configurations.
Shifted Kβ-stability is defined through a positive uniform lower bound, Fβ(T)≥δJ(T), for every Kähler test configuration. The best bound is then optimized over β to define the threshold σ(X,α).
The main theorem connects three conditions in the stated compact Kähler setting: a positive scalar curvature metric exists, shifted Kβ-stability holds for some β, and σ(X,α) is positive. The paper presents these conditions as equivalent.
Two matching views in the polarized setting
In the polarized setting, the preprint identifies the transcendental positive-scalar-curvature threshold with its non-Archimedean counterpart. The two thresholds therefore coincide within the setting covered by the result.
The same setting also gives an equivalence between the existence of a positive scalar curvature Kähler metric and shifted K-stability. To prove the converse, the paper uses approximations of finite-energy geodesic rays by rays from Kähler test configurations, together with continuity of the associated radial functionals.
An entropy-regularization argument identifies the infimum of the Mna/Jna ratio over Hna with the corresponding finite-energy infimum and with σ−s, using the paper’s notation for the threshold comparison.
The boundary between the regimes
The common threshold separates four regimes. When σ≤0, no positive scalar curvature metric exists. When 0<σ<s, a positive scalar curvature metric exists without a constant scalar curvature Kähler metric. At σ=s, the threshold alone leaves the cscK question unresolved. When σ>s, a unique cscK metric exists.
The comparison with s also classifies classical stability. The paper states that σ≥s corresponds to K-semistability, σ>s to uniform K-stability, and σ<s to K-instability.
The critical case is the main qualification to the threshold test. At σ=s, cscK existence depends on the zero set of the non-Archimedean functional Mna and is not determined by the numerical threshold alone.
A theorem with a defined scope
The conclusions are stated for compact Kähler manifolds and polarized settings under the assumption of positive average scalar curvature. The threshold equality and polarized shifted-stability result are limited to the polarized setting specified by the paper.
That scope matters when interpreting the result: the supplied analysis does not extend the criterion beyond the positive-average-scalar-curvature assumption, and the numerical threshold by itself does not settle the cscK question at σ=s.
The author reports using ChatGPT 5.6 Sol for preliminary brainstorming, language editing and consistency checks, while stating that the mathematical statements, proofs and citations were independently checked. The acknowledgment reports encouragement and helpful discussions with colleagues but does not identify a funding source.
Paper data and sources
Original title: Positive scalar curvature Kähler metrics and shifted $K$-stability
Authors: Zehao Sha
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text