The awkward title is an attempt at approximating the following specific question: Let $(M^{2n}, J)$ be a complex manifold, suppose $g_0$ is a Riemannian metric $M$ compatible with $J$, and suppose $\phi_t$ is a one parameter family of diffeomorphisms of $M$ generated by a vector field $X$ such that $\phi_t^* g$ is compatible with $J$ for all $t$. The questions is: must $X$ be a (real part of a) holomorphic vector field in the sense that $L_X J = 0$?
I can give a proof that yes, it is true, in the caseAssume that $g$$g_0$ is also Kahler, by showing that one has and that $X = \nabla f$ for some smooth function $f$. As the family of Kahler forms $\omega_t = (\phi_t^* g)(\cdot, J \cdot)$ satisfies $\omega_t = \phi_t^* \omega_0$, after which one obtains$\phi_t^* g_0$ remains compatible with $\phi_t^* J = J$$J$, and the claimit follows. In general the hypotheses imply that that $L_X g$$L_{\frac{1}{2} \nabla f} g_0 = \nabla^2 f$ is of type $(1,1)$, and. Using the claim seems equivalent to showingK"ahler hypothesis one can show that $L_X \omega$ is also of type$(\nabla^2 f)^{1,1}(\cdot, J \cdot) = \frac{1}{2} d d^c f$. On the other hand, since the associated K"ahler form $(1,1)$$\omega$ is closed, but I can only show this withit follows from the Kahler hypothesisCartan formula that $L_{\frac{1}{2} \nabla f} \omega = \frac{1}{2} d d^c f$. Thus it follows that $L_{\frac{1}{2} \nabla f} \omega = (L_{\frac{1}{2} \nabla f} g)(\cdot, J \cdot) = L_{\frac{1}{2} \nabla f} \omega + g(L_{\frac{1}{2} \nabla f} J\cdot , \cdot)$, and hence $L_{\nabla f} J \equiv 0$, as required.
