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.



Symplectic form on a Kähler manifold can be not real analytic?
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 non-analytic $2$-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.


