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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 theorem. My 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.

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See the answers to this question mathoverflow.net/questions/96910/… –  Paul Reynolds Jun 2 '12 at 23:38
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As far as I can tell Robert Bryants answer to that question just contains a definition of what one should call a harmonic morphism, i.e. the obvious one and the reference he gives discusses only Riemannian manifolds. –  orbifold Jun 3 '12 at 0:08
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The Baird-Wood book "Harmonic Morphisms Between Riemannian Manifolds" in that bibliography, has a full chapter (Chapter 8) on Holomorphic harmonic morphisms. –  Francois Ziegler Jun 3 '12 at 1:05
    
Beware that the property of being "harmonic" depends on the metrics involved, so we are really talking about properties of Kahler manifolds $(X,\omega)$ and $(Y,\alpha)$. The obvious condition to demand for anything like pullbacks being harmonic to hold is that $f^*\alpha = \omega$, but this severely limits the canditates for $f$; I have a hard time thinking of other things than immersions and finite morphisms covering that could verify this (since no morphism w/positive dimensional fiber can). –  Gunnar Magnusson Jun 3 '12 at 9:58
    
*finite morphisms covering $\to$ finite coverings –  Gunnar Magnusson Jun 3 '12 at 9:59

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