12 problems

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

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…

Generalized Kähler–Ricci-flow convergence conjecture. The generalized Kähler–Ricci flow preserves the generalized Kähler class, exists for all time, and converges to the unique gen…

Generalized Kähler Calabi–Yau conjecture. There exists a unique generalized Calabi–Yau geometry $(g_{CY},b_{CY},I,J_{CY})\in[(g,b,I,J)]$, and furthermore $(g_{CY},I)$ and…

Contractibility conjecture. The space of canonical deformations of a fixed generalized Kähler structure is contractible.

Let $(g, I, J)$ be a generalized Kähler structure of symplectic type with toric symmetry, and let $J_{\infty}$ denote the limiting complex structure obtained from a convergent sequ…

Commuting generalized Kähler-flow regularity conjecture. The solution to generalized Kähler-Ricci flow with initial condition $g_0$ exists on $[0,\tau^*(g_0))$. This is presented a…

Let $(M^{4n}, g, I, J)$ be a nondegenerate generalized Kähler structure, and let the generalized Kähler Ricci flow start from this structure. Generalized Kähler Ricci flow conjectu…

Let $(M^{4n}, g, I, J)$ be a nondegenerate generalized Kähler manifold, and consider its $\Omega$-Hamiltonian deformation class. Generalized Kähler Calabi–Yau conjecture. There exi…

Let $(M^{2n},g,J_\pm)$ be a generalized Kähler structure satisfying $[J_+,J_-]=0$. Let $\mathcal H$ be the space of formally generalized Kähler classes, and let $P$ be the associat…

Let $(M^{2n},g_0,J_{\pm})$ be a generalized Kähler manifold with $[J_+,J_-]=0$. Let $\tau^*$ be the supremum … where $\mathcal P$ is the positive cone of formally generalized Kähle…

Generalized-potential conjecture. Many of the properties of GKG, such as the existence of a generalized potential, should hold for arbitrary-signature bi-hermitian geometries.