Let $(X,\omega)$ be an $n$-dimensional Gauduchon manifold, i.e. $\partial \bar{\partial}\omega^{n-1}=0.$ Let $Y$ be a $p$-dimensional complex sub-manifold of $X$ with the K$\ddot{a}$hler $(1,1)$-form $\omega_Y=\omega|_Y,$where $p $\partial \bar{\partial}\omega_Y^{p-1}=0?$ I appreciate any comment. Thank you very much!
Complex submanifolds of Gauduchon manifolds
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differential-geometry
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1Why is $\omega_Y$ a Kähler form on $Y$? – 2017-02-27
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0Sorry, I do not express it explicitly. Since $Y$ is a complex submanifold of X, there is an inclusion map $i : Y \hookrightarrow X.$ Set $\omega_Y:=i^*\omega$. – 2017-02-28
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0Of course. But I thought a Gauduchon metric was not a Kähler metric. So perhaps you did not mean to call $\omega_Y$ a Kähler form on $Y$? – 2017-02-28
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0Yes,you are right, $\omega_Y$ may not be Kahler.. However, I do not know how to show $\omega_Y$ is Gauduchon, that is how to show $\partial \bar{\partial}\omega_Y^{p-1}=0.$ Can you present the proof of it?Thank you! – 2017-02-28