Calabi–Yau threefold symplectic Monge–Ampère conjecture

Let $M$ be a compact Kähler manifold of complex dimension $n=3$ with $c_1(M)=0$. Let $\omega$ be a Kähler form and let $\Omega=\rho+\sqrt{-1}\sigma$ be a holomorphic volume form. For a function $F:M\to\mathbb{R}$ satisfying

$$\int_M e^F\omega^n=\int_M\omega^n,$$

consider the equation

$$(\Omega-\sqrt{-1}dd^s\varphi\Omega)\wedge(\bar{\Omega}+\sqrt{-1}dd^s\varphi\bar{\Omega})=e^F\Omega\wedge\bar{\Omega},$$

or equivalently

$$(\rho+dd^s\varphi\sigma)\wedge(\sigma-dd^s\varphi\rho)=e^F\rho\wedge\sigma.$$

Calabi–Yau threefold symplectic Monge–Ampère conjecture. The equation has a unique solution $\varphi:M\to\mathbb{R}$, modulo adding a constant, such that $\tilde{\rho}=\rho+dd^s\varphi\sigma$ is positive, $\tilde{\sigma}=\sigma-dd^s\varphi\rho$ is negative, and $\tilde{\rho}$ is dual to $\tilde{\sigma}$.

The conjecture is an analog of Calabi's conjecture in which the holomorphic volume form is deformed while the Kähler form is fixed. The paper states the relevant notions of positivity and duality later; no resolution of this conjecture is supplied in the given text.