Pullback of harmonic forms. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T08:58:15Z http://mathoverflow.net/feeds/question/98690 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/98690/pullback-of-harmonic-forms Pullback of harmonic forms. orbifold 2012-06-02T23:25:50Z 2012-06-03T00:09:19Z <p>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:</p> <p>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?</p> <p>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. </p>