Let $X$ be a projective variaty which blow up at a point $p$ , i.e, $X=Bl\_p(\mathbb CP^2)$, then for the Line bundle $L=-K\_X$, we have for [Futaki invariant](https://web.math.princeton.edu/~chil/Futaki.pdf) $Fut\_L\neq 0$, I want to see, what about other Line bundles, have we same result?



Futaki invariant on $X=Bl_p(\mathbb CP^2)$ for different line bundles
My knowledge is by no means up-to-date, but in case it's useful for posterity, the situation for cohomogeneity-one almost-homogeneous spaces with two ends (which includes Hirzebruch surfaces, the blow-up of $\mathbf{P}^{n}$ along linear subspaces of complementary dimension, etc.) can be found in [*On existence of Kähler metrics with constant scalar curvature*](http://ir.library.osaka-u.ac.jp/dspace/bitstream/11094/7621/1/ojm31_03_05.pdf), Osaka J. Math., **31** (1994), 561–595.


Briefly, on a compact almost-homogeneous Kähler manifold $X$ with real hypersurface orbits and two ends, there exist Kähler classes with vanishing Futaki character if and only if the connected automorphism group is reductive.


Incidentally, such an $X$ is a compactification of a holomorphic $\mathbf{C}^{\times}$-bundle $\Lambda$ over a compact, homogeneous Kähler manifold $M$ by a classification of Huckleberry and Snow. The value of the Futaki character on the Euler vector field of $\Lambda$ is an explicit one-variable integral depending on real parameters defining the Kähler class and integers specifying the Chern class of $\Lambda$.


