let $(M,J,g,\omega)$ be a compact K\"ahler manifold of complex dimension at least $2$. As usual $J$ is the complex structure, $\omega$ is the symplectic form, $g$ is the Riemannian metric and 
$$\omega(\cdot,\cdot)=g(J\cdot, \cdot)\,.$$


I denote with $Ham(M,\omega)$ the group of Hamiltonian symplectomorphisms w.r.t. the symplectic form $\omega$ and with $Iso\_{0}(M,g)$ the identity component of the group of isometries w.r.t. the metric $g$.


Is the group $Ham(M,\omega)\cap Iso\_{0}(M,g)$ compact? Maybe it is an easy question but i haven't figured out the answer yet. Any suggestion or hint is welcome and if the question is not well suited for this site i'll move it to MSE.


Thank you in advance. 



Is the group $Ham(M,\omega)\cap Iso_{0}(M,g)$ compact?
By Ascola-Arzelà theorem, the group of isometries of a compact metric space is itself compact and then a compact connected Lie Group. The condition that the isometries preserve the complex structure is a closed condition, so your subgroup is compact.


