Let $M$ be a Kähler manifold. The complex structure on it naturally gives rise to the real analytic structure. I wonder if there exist Kähler manifolds such that the associated symplectic $2$form $\omega$ is $C^\infty$smooth but not real analytic.

1$\begingroup$ What about a small disk in $\mathbb{C}$ with $\omega = \overline{\partial}\partial \left(\z\^2 + \varepsilon e^{1/z}\right)$? $\endgroup$ – Bertram Arnold Dec 1 '18 at 14:00

$\begingroup$ @BertramArnold Yes, you're right but it should be $e^{1/z^2}$ instead of $e^{1/z}$. Just a misprint. $\endgroup$ – anna abasheva Dec 2 '18 at 13:49
The answer is positive for any Kähler manifold.
Consider first surfaces. Take a compact Riemann surface $\Sigma$, then any symplectic form on it is associated to a Kähler form, so the answer is yes, since there are plenty of nonanalytic $2$forms.
More generally, for any Kähler manifold $M$ take any Kähler form $\omega$ and let $\omega_1$ be a closed $(1,1)$form supported in a ball $B\subset M$. Then $\omega+\varepsilon \omega_1$ is Kähler for small $\varepsilon$, but not analytic.

$\begingroup$ Why doesn't the second argument work for a Riemann surface? $\endgroup$ – Deane Yang Dec 2 '18 at 2:53

1$\begingroup$ Dear Deane, I have not said that it doesn't work for surfaces, it does. $\endgroup$ – Dmitri Panov Dec 2 '18 at 11:11
Not quite an answer, but something related to it.
There is this paper where the authors prove, that for every symplectic manifold $(M,\omega)$ there is an analytical manifold $M^a$ and an analytical symplectic form $\omega^a$ such that $(M,\omega)$ and $(M^a,\omega^a)$ are symplectomorphic.
Edit: The manifold $(M^a,\omega^a)$ is unique up to symplectomorphisms.