1
$\begingroup$

Good time of day. I have the following question.

$H$ - Hopf surface i.e. quotient $\mathbb{C}^2 \setminus \{ 0 \}$ by the action of $\mathbb Z$, where the action of $k\in \mathbb Z$ is given by $z \to \lambda^{k} z$ for some $\lambda>1$. $\widetilde H$ is the blow-up of $H$ at a point. Does admit $\widetilde H$ a Kahler metric?

It's known that there is no Kahler metric in $H$. But $H$ has local conformal Kahler metric (l.c.K.). And from the famous theorem "The blow-up at a point of a l.c.K. manifold admits a l.c.K. structure" we obtain that $\widetilde H$ has also local conformal Kahler metric.

I don't know what's going on with global Kahler metric on $\widetilde H$. Is there exist Kahler metric or not? Please help me. If you don't mind please explain it in more details. Thank you

Improve this question
$\endgroup$
4
  • 5
    $\begingroup$ If you blow up a Hopf surface at a point, the first Betti number is still one. So it can't be Kähler. $\endgroup$
    – Donu Arapura
    Commented May 18, 2022 at 22:08
  • 4
    $\begingroup$ It cannot even be symplectic, since there's no class in $H^2$ that has positive square. $\endgroup$
    – Marco Golla
    Commented May 18, 2022 at 22:12
  • $\begingroup$ @Donu Arapura thank you for your answer. How to prove that the first Betti number preserves? Is it truth in general case for any compact complex manifold or not? $\endgroup$
    – UserIn
    Commented May 18, 2022 at 22:15
  • 4
    $\begingroup$ You can compute the homology of a blow up as in pp 473-474 of Griffiths-Harris, and check that $H_1$ doesn't change. $\endgroup$
    – Donu Arapura
    Commented May 18, 2022 at 22:30

0

Reset to default

You must log in to answer this question.