Suppose we have two Kähler manifolds $(M, w_1, J_1)$ and $(N, w_2,J_2)$ and a diffeomorphism $f:M \rightarrow N$ such that $f$ preserves the complex structure and the symplectic form.
If $\Omega$ is a holomorphic $(n,0)$-form on $N$, when can we say the pullback of $\Omega$ is a holomorphic $(n,0)$-form on $M$?
If $f : M \to N$ preserves the complex structure, that is $df\circ J_1 = J_2\circ df$, then $f$ is a holomorphic map. As holomorphic maps preserve the bidegree of a complex differential form, if $\alpha \in \mathcal{E}^{p,q}(N)$, $f^*\alpha \in \mathcal{E}^{p,q}(M)$. Furthermore, $\bar{\partial}$ commutes with pullback by holomorphic maps, i.e. $\bar{\partial}(f^*\alpha) = f^*\bar{\partial}\alpha$. So if $\alpha$ is a holomorphic $(p, 0)$-form on $N$, $f^*\alpha$ is a holomorphic $(p, 0)$-form on $M$.
Note, the conclusion does not require the complex structures to be Kähler, the map to preserve the symplectic forms, or for the map to be a diffeomorphism. All that is needed is that the map is holomorphic.