Yau Stability Conjecture for Constant Scalar Curvature Metrics

Definitions and mathematical statement

Let $(X,L)$ be a polarized compact Kähler manifold, with $L$ an ample holomorphic line bundle, and let $[\omega_0]$ be the Kähler class in the source. A representative $\omega\in[\omega_0]$ has constant scalar curvature if $\operatorname{Scal}(\omega)$ is constant on $X$.

The source asks, as a conjecture attributed to Yau, whether the following equivalence holds: \[ \exists\,\omega\in[\omega_0]\text{ Kähler with constant scalar curvature} \quad\Longleftrightarrow\quad L\to X\text{ is stable in the intended geometric invariant theory sense}. \] The right-hand side is retained in the source's wording: it does not specify a GIT parameter space, a linearized group action, an asymptotic embedding criterion, or a definition of K-stability. Consequently strict stability, polystability, and their uniform or relative versions cannot be interchanged here. The relationship between $[\omega_0]$ and $c_1(L)$, including normalization, must also be fixed; the supplied statement does not explicitly give it. No hypothesis on the automorphism group of $(X,L)$ is stated. These choices are necessary to turn the displayed source question into a fully defined algebraic equivalence.

Short English statement

Ask whether a polarized Kähler class contains a constant scalar curvature metric exactly when the polarization satisfies the intended algebraic stability condition. The source leaves the stability notion, automorphism qualifications, and class normalization unresolved.

Sources

• Local UnsolvedMath registry, permanent ID $30001003$, source code catalogue entry, "Yau Stability Conjecture for Constant Scalar Curvature Metrics"; the conjecture is attributed to Yau in the original. Registry curation: Ulam AI / UnsolvedMath Contributors, CC BY 4.0; underlying sources retain their supplied provenance.
• Real Analysis, Harmonic Analysis and Applications (2009), Oberwolfach Reports, pp. 1771--1850, DOI https://doi.org/10.4171/owr/2008/32.
• Supplied "Chen--Cheng cscK theorem": https://arxiv.org/abs/1801.05907.
• Supplied "current K-stability survey": https://doi.org/10.1017/fms.2023.104.
• Supplied literature triage, literature_checked_at: 2026-08-21: labeled partially solved; says major cases and analytic directions of the cscK--stability correspondence are proved, while the general algebraic Yau--Tian--Donaldson equivalence in the stated form remains unresolved. This is dated registry curation requiring independent status and hypothesis review.