Let $X$ be a smooth projective variety over $\mathbb{C}$. Consider the Kahler structure $(J,g,\omega)$ on $X$ induced by the Fubini-Study metric. Let $Symp(X,\omega)$ be the group of symplectomorphisms of $(X,\omega)$.


**QUESTION.**


Is there a finite order symplectomorphism $f \in Symp(M,\omega)$ which is not conjugate (in $Symp(X,\omega)$) to an algebraic automorphism ?


I am just looking for one example, preferably such that the dimension of $X$ is as small as possible.


EDIT


How about if we require $f$ to have no fixed points?



Example of finite order symplectomorphism which is not an automorphism
Here is one example - take an elliptic curve without an automorphism of order $4$ *with a fixed point*. Note at the same time that any torus $T^2$ with an area form has an area preserving automorphism of order $4$ *with a fixed point*. 


