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Let $(M, \omega_M, J_M)$ and $(N, \omega_N, J_N)$ be compact Kähler manifolds. Denote $g_M=\omega_M(\cdot, J_M\cdot)$ and $g_N=\omega_N(\cdot, J_N\cdot)$.

Assume there is a diffeomorphism $\nu:M\to N$ such that $\nu^*(\omega_N)=\omega_M$, there is a diffeomorphism $\phi:M\to N$ such that $\phi^*(J_N)=J_M$ and there is a diffeomorphism $\chi:M\to N$ such that $\chi^*g_N=g_M$.

Is there a diffeomorphism $\psi:M\to N$ such that $\psi^*(\omega_N)=\omega_M$ and $\psi^*(J_N)=J_M$ (and hence also $\psi^*(g_N)=g_M$)?

Let $(M, \omega_M, J_M)$ and $(N, \omega_N, J_N)$ be compact Kähler manifolds. Denote $g_M=\omega_M(\cdot, J_M\cdot)$ and $g_N=\omega_N(\cdot, J_N\cdot)$.

Assume there is a diffeomorphism $\nu:M\to N$ such that $\nu^*(\omega_N)=\omega_M$, there is a diffeomorphism $\phi:M\to N$ such that $\phi^*(J_N)=J_M$ and there is an orientation-preserving diffeomorphism $\chi:M\to N$ such that $\chi^*g_N=g_M$.

Is there a diffeomorphism $\psi:M\to N$ such that $\psi^*(\omega_N)=\omega_M$ and $\psi^*(J_N)=J_M$ (and hence also $\psi^*(g_N)=g_M$)?