6 problems

Poisson vanishing conjecture. The Hodge numbers $h^{2j,0}(\mathsf{X})$ are nonzero for every $j\leq\tfrac{d}{2}$. In particular, if $h^{2,0}(\mathsf{X})=0$, then every holomorphic…

Global Weinstein splitting conjecture. There exist a holomorphic symplectic manifold $\mathsf{Y}$ of dimension $d$, a holomorphic Poisson manifold $\mathsf{Z}$, and a holomorphic P…

Cohomological Kähler foliation conjecture. Every regular foliation on a compact Kähler manifold is cohomologically Kähler.

Let $(X,\Lambda_0)$ and $(Y,\Pi_0)$ be compact holomorphic Poisson manifolds, let $f:(X,\Lambda_0)\to(Y,\Pi_0)$ be a holomorphic Poisson map, and let … be a Poisson analytic family…

Kuranishi-type completeness conjecture. A complete family of deformations of $(X,\Lambda_0)$ whose Poisson Kodaira–Spencer map is an isomorphism exists without assuming…

Theorem of Completeness for holomorphic Poisson structures. If $\varphi_0$ is surjective, then the Poisson analytic family $(\mathcal{X},\Lambda,B,\pi)$ is complete at $0\in B$. Th…