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Consider $\mathbb{C}^{2}$ equipped with the Kähler form $$ \omega_{\mu}=\frac{i}{2} \partial \bar{\partial} \log \left(\left(1+|z|^{2}\right)^{\mu}+|w|^{2}\right), $$ where $\mu$ is a positive real number.

Question. Does there exists a complete Kähler manifold $\left(M, \hat{\omega}_{\mu}\right)$ such that

  1. $\mathbb{C}^{2} \subset M$,
  2. $\mathbb{C}^{2}$ is dense in $M$,
  3. $\hat{\omega}_{\left.\mu\right|_{\mathbb{C} 2}}=\omega_{\mu} ?$

Remark. The complex projective plane $M=\mathbb C P^{2}$ equipped with the Fubini-Study form is a positive answer to the question, when $\mu=1 .$ I suspect this is the only case but I am not able to prove it.

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  • $\begingroup$ This is an interesting problem, what is the motivation? $\endgroup$
    – AmorFati
    Commented Jul 29, 2020 at 8:00
  • $\begingroup$ I think that if mu is not one, then the metric completion adds only two points, so it's not a manifold $\endgroup$
    – Martin de Borbon
    Commented Jul 29, 2020 at 8:24
  • $\begingroup$ @MartindeBorbon Thank you for the answer! But I did not understand your argument, could you elaborate it, please? $\endgroup$
    – Robbixmaths
    Commented Jul 29, 2020 at 12:24
  • $\begingroup$ If you restrict to z=0 you add a point and get a sphere. If you restrict to w=0 you add another point and get a sphere (scaled by mu). I think, if mu is not 1, those are the only two points. Let me restrict to mu>1. Look at the distance from $(z_0, w_0)$ to $(z_0, 0)$ where $(z_0, w_0)$ belongs to the line $z= \lambda w$ for some fixed nonzero $\lambda$. Restrict to the line $z=z_0$, you get a sphere and the distance squared from $(z_0, w_0)$ to $(z_0, 0)$ is roughly $|z_0|^{1-\mu}$ so goes to zero as $z_0$ goes to infinity $\endgroup$
    – Martin de Borbon
    Commented Jul 29, 2020 at 13:50
  • $\begingroup$ Btw, your metric is toric, so one could look at the moment polytope and see it does not corresponds to a manifold $\endgroup$
    – Martin de Borbon
    Commented Jul 29, 2020 at 13:53

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