The Yau–Tian–Donaldson Conjecture for Constant Scalar Curvature Kähler Metrics

The Yau–Tian–Donaldson conjecture predicts that a polarized manifold carries a canonical Kähler metric in its polarization class exactly when it is K-polystable. Settled for Kähler–Einstein metrics on Fano manifolds, a proof of the uniform version of such a conjecture for general constant scalar curvature Kähler metrics has been showed in a recent preprint (arxiv:2605.30063v2). While the original, non-uniform Yau-Tian-Donaldson conjecture has been proved to be false: there is a polarized smooth projective fivefold that is K-polystable but admits no extremal Kähler metric in $c_1(A)$, so K-polystability does not imply existence. This makes the uniform version of the Yau-Tian-Donaldson Conjecture optimal as experts expected.

The paper's appendix draws a distinction worth keeping: a counterexample may reduce to a finite certificate, checkable once the object is written down, or it may itself be a theorem quantified over all degenerations. This is the second kind. The method field records construction, because the resolution exhibits an explicit fivefold, but the difficulty lay elsewhere - candidate manifolds of this shape have been available since 2008, and what was missing was the proof that the mechanism works.