Does there exist a closed connected smooth manifold that admits exactly one (up to biholomorphism) integrable complex structure and exactly one (up to symplectomorphism and rescaling) symplectic structure which are not compatible with each other (i.e. do not define Kaehler structure)?

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    $\begingroup$ I don't know, but I have a remark. For the most famous non-Kahler symplect and complex manifold, the Kodaira Thurston manifold, there is not a unique symplectic form since Thurstons construction allows us to change the area of the fibre and base of the $\mathbb{T}^{2}$-fibration over $\mathbb{T}^{2}$, so there is a two dimension family of symplectic forms at least. The other examples I know of such manifolds come from nil-geometry I would imagine that some similar argument could show that there are many symplectic forms. $\endgroup$ – Nick L Jul 12 at 6:44
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    $\begingroup$ Also, I think that sym. manifolds with a unique symplectic structure in this sense are quite rare. It is a famous result of Gromov that $\mathbb{CP}^{2}$ has unique symplectic form in this sense. But I don't think we are able to establish any uniqueness results in higher dimensions presently. Maybe it is worth to look at Salomons survey paper "uniqueness of symplectic structures" for some more information. $\endgroup$ – Nick L Jul 12 at 6:47

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