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 $Fut_L\neq 0$, I want to see, what about other Line bundles, have we same result?
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1$\begingroup$ What is the Futaki invariant? $\endgroup$– Piotr AchingerDec 22, 2014 at 2:38
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$\begingroup$ @PiotrAchinger, ohh thanks, I added a refference for the definition of Futaki invariant $\endgroup$– user21574Dec 22, 2014 at 4:08
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$\begingroup$ Do you mean "other Kähler classes" (on $X$) rather than "other line bundles"? $\endgroup$– Andrew D. HwangDec 22, 2014 at 17:55
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$\begingroup$ I mean on $\mathcal H_L=\{\text{kahler forms} \omega \text{on L which} [\omega]=c_1(L)\}$ $\endgroup$– user21574Dec 22, 2014 at 18:02
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1$\begingroup$ 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. $\endgroup$– Andrew D. HwangDec 22, 2014 at 18:24
1 Answer
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, 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 $\Lambda$.