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2d
answered Futaki invariant on $X=Bl_p(\mathbb CP^2)$ for different line bundles
2d
comment 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 non-vanishing. 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 non-reductive automorphism group, so it admits no Kähler metric of constant scalar curvature.
2d
comment 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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