$\require{AMScd}$

**Preliminaries:** Let $(X,\omega,J)$ be a closed Kahler manifold. That is, $X$ is a closed $2n$-manifold, $\omega$ is a symplectic form and $J$ is a compatible (integrable) complex structure. Suppose that $[\omega] = \lambda \cdot c_1(X,J)$ with $\lambda \in \mathbb{R}$, where $[\omega] \in H^2(X)$ is the de Rham class of $\omega$ and $c_1(X,J)$ is the 1st Chern class of $X$.

When $\lambda \le 0$, famous theorems of Yau and Aubin (in the $\lambda < 0$ case) state that there exists a *unique* Kahler (symplectic) form $\omega_{KE}$ on $X$ that is compatible with $J$ and that satisfies the Kahler-Einstein condition $\omega_{KE} = \lambda \cdot \text{Ric}[\omega_{KE}]$. Here $\text{Ric}[\omega_{KE}]$ is the Ricci form.

**Main Questions:** My question is essentially: how unique is a Kahler-Einstein complex structure if it is compatible with a fixed symplectic form? More precisely, let $(X,\omega)$ be a closed symplectic manifold and let $J$ be a compatible complex structure with respect to which $\omega$ is Kahler-Einstein.

Question 1:Is $J$ unique in the sense that, for any other compatible Kahler-Einstein $J'$ on $(X,\omega)$, there is a biholomorphic symplectomorphism $\varphi:X \to X$, i.e. a diffeomorphism $\varphi$ such that $\varphi^*\omega = \omega$ and $T\varphi \circ J = J' \circ T\varphi$?

I would also be interested in the following restricted cases of Question 1.

Question 2:Is Question 1 true if we restrict to either the case where $[\omega] = -c_1(X,J)$? What if we assume $\text{dim}_{\mathbb{C}}(X) = 2$? What if we assume both?

Finally, I would be interested in other curvature conditions guaranteeing a uniqueness property in the spirit of Question 1. For instance, we could ask the following.

Question 3:Let $(X,\omega,J)$ be a Kahler manifold with constant holomorphic sectional curvature. Then is $J$ unique in the sense of Question 1?