1
$\begingroup$

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?

$\endgroup$
5
  • 1
    $\begingroup$ What is the Futaki invariant? $\endgroup$ Dec 22, 2014 at 2:38
  • $\begingroup$ @PiotrAchinger, ohh thanks, I added a refference for the definition of Futaki invariant $\endgroup$
    – user21574
    Dec 22, 2014 at 4:08
  • $\begingroup$ Do you mean "other Kähler classes" (on $X$) rather than "other line bundles"? $\endgroup$ Dec 22, 2014 at 17:55
  • $\begingroup$ I mean on $\mathcal H_L=\{\text{kahler forms} \omega \text{on L which} [\omega]=c_1(L)\}$ $\endgroup$
    – user21574
    Dec 22, 2014 at 18:02
  • 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. $\endgroup$ Dec 22, 2014 at 18:24

1 Answer 1

2
$\begingroup$

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 $\Lambda$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.