Generalized Kähler Calabi–Yau conjecture

Let $(M^{2n},g,b,I,J)$ be a compact generalized Kähler manifold with holomorphically trivial canonical bundles. A generalized Calabi–Yau geometry is a generalized Kähler structure determined by global closed pure spinors $\psi_1$ and $\psi_2$ for which

$$\Phi=-\log\frac{(\psi_1,\overline{\psi}_1)}{(\psi_2,\overline{\psi}_2)}\equiv\lambda,$$

where $(\cdot,\cdot)$ is the Mukai pairing and $\lambda$ is a constant. The generalized Kähler class of $(g,b,I,J)$ is denoted by $[(g,b,I,J)]$.

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 $(g_{CY},J_{CY})$ are both Kähler Ricci-flat.

This is the generalized Kähler analogue of the Calabi conjecture, replacing a scalar potential equation by a problem in the nonlinear space of generalized Kähler structures. The paper develops generalized Kähler–Ricci-flow estimates supporting the conjecture, but the general existence and uniqueness assertion remains open.