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answered  Futaki invariant on $X=Bl_p(\mathbb CP^2)$ for different line bundles 
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Futaki invariant on $X=Bl_p(\mathbb CP^2)$ for different line bundles
If I understand your question: the Futaki character for an arbitrary line bundle (a.k.a., integral Kähler class) on this particular Hirzebruch surface $X$ is nonvanishing. Indeed, every Kähler class on $X$ admits an extremal metric. If the Futaki character vanished for some class, the corresponding extremal metric would have constant scalar curvature. But $X$ has nonreductive automorphism group, so it admits no Kähler metric of constant scalar curvature. 
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Futaki invariant on $X=Bl_p(\mathbb CP^2)$ for different line bundles
Do you mean "other Kähler classes" (on $X$) rather than "other line bundles"? 
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What is a branched Riemann surface with cuts?
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