Let $X$ and $Y$ be two smooth complex projective varieties. So in particular they are Kähler manifolds, and hence we can consider them as algebraic varieties as well as symplectic manifolds.
If $f:X \to Y$ is an isomorphism in the category of algebraic varieties, must it be an isomorphism in the category of symplectic manifolds (i.e., both $f$ and $f^{-1}$ are symplectomorphisms)? I think the opposite is false.