Poisson K-polystability and the semiclassical Yau–Tian–Donaldson correspondence

Let $(X,\sigma,\alpha)$ be a compact Kähler holomorphic Poisson manifold, where $\sigma$ is its holomorphic Poisson tensor and $\alpha$ is its Kähler class. A Poisson K-polystable triple is one satisfying the Poisson K-polystability condition introduced in the paper. A cscGK structure is a constant-scalar-curvature symplectic generalized Kähler structure.

Semiclassical Yau–Tian–Donaldson conjecture. The triple $(X,\sigma,\alpha)$ is Poisson K-polystable if and only if there exists $\epsilon>0$ such that, for every $\lambda\in\mathbb C$ with $|\lambda|<\epsilon$, the triple $(X,\lambda\sigma,\alpha)$ admits a cscGK structure.

This conjecture proposes a small-Poisson-tensor extension of the Yau–Tian–Donaldson correspondence, relating Poisson K-polystability to the existence of generalized Kähler metrics of constant scalar curvature. The supplied text does not state a resolution; it notes that the corresponding assertion has been confirmed in the toric setting after replacing Poisson K-polystability by a suitable uniform stability condition.