The polarized semiclassical Yau–Tian–Donaldson conjecture for holomorphic Poisson tensors

Let $(X,L)$ be a smooth polarized projective manifold admitting a cscK metric in the class $2\pi c_1(L)$, and let $\sigma\in H^0(X,\wedge^2T^{1,0}_X)$ be a non-zero holomorphic Poisson tensor. The group $\operatorname{Aut}_{\rm red}(X)$ acts linearly on $H^0(X,\wedge^2T^{1,0}_X)$, and $\sigma$ is polystable if it is a polystable point for this action.

Polarized semiclassical Yau–Tian–Donaldson conjecture. The following are equivalent:

1. The bivector $\sigma$ is polystable for the linear action of $\operatorname{Aut}_{\rm red}(X)$ on $H^0(X,\wedge^2T^{1,0}_X)$.
2. There exists $\epsilon>0$ such that, for $|\lambda|<\epsilon$, $(X,\lambda\sigma,2\pi c_1(L))$ admits a cscGK structure.

This is the polarized projective ramification of the semiclassical Yau–Tian–Donaldson conjecture. It specializes Poisson K-polystability to geometric invariant theory polystability when the underlying polarized manifold already admits a cscK metric. The supplied text gives no resolution, so the equivalence remains open here.