$\require{AMScd}$


**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.



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> **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 $T\varphi \circ J = J' \circ T\varphi$?
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I would also be interested in the following restricted cases of Question 1.



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> **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?
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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.



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> **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?
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Uniqueness of a compatible Kahler-Einstein structure on a symplectic manifold?
Concerning 2) one can, of course, take the product of two curves of higher genus to get a counter-example. In general, to have a statement as you want, one should look for rigid complex surfaces of general type (with ample $K$), because as soon as such a surface has a deformation, we get a counterexample (by Yau Aubin). 


Concerning 3), the situation depends on whether the curvature is zero or not. The universal cover of a manifold of constant holomorphic sectional curvature is either i) $\mathbb CP^n$ with Fubini-Study metric, or ii) $\mathbb C^n$ with a flat metric or iii) the open unit ball $\mathbb B^n$ with complex hyperbolic metric. See the discussion here (don't be mislead by the title :) ):


[Kahler manifolds with constant bisectional curvature](https://mathoverflow.net/questions/105687/kahler-manifolds-with-constant-bisectional-curvature)


In the case i) it is a well known theorem, that a Kaehler manifold diffeomorphic to $\mathbb CP^n$ is in fact biholomoprhic to $\mathbb CP^n$. So the answer to your question is positive.


In the case ii) we can take two elliptic curves with a flat metric of area $0$, that are not biholomorphic. So the answer to the question is negative. 


Finally, I believe that the answer is also negative in case iii), provided the dimension is greater than $1$. This is certainly true for complex dimension $2$. Indeed, by Yau's theorem any complex surface diffeomorphic to a ball quotient is in fact biholomorphic to it, and also the complex structure on a ball quotient is rigid.


