If $f \colon X \to Y$ is a holomorphic map between Kaehler manifolds, then the pullback of a harmonic form on $Y$ is not necessarily harmonic on $X$, even if $f$ is an immersion. This came up during a discussion of the hodge Hodge theorem. My question questions are, mainly out of curiosity:
Is there a nice characterization of maps for which the pullback is harmonic? Can one say anything interesting about the harmonic projection of the pullback?
Since I expect that a full answer to the first question is unknown, I am of course happy with partial results. I am also dimly aware of "harmonic maps", i.e. critical points of $S[\phi] = \int \langle d\phi \wedge \star d\phi\rangle$, however I am not sure under which conditions they are holomorphic and I was told that they are not closed under composition.