4
$\begingroup$

Let $X$ be a noncompact complex manifold which contains no positive dimensional compact analytic sets.

Conjecture: There must be strictly plurisubharmonic functions on $X$ .

Is it true?

Improve this question
$\endgroup$
1
  • 5
    $\begingroup$ Is it true for $X$ being the blow-up of $\mathbb{C}^2$ at the origin minus a point in the fiber $\mathbb{C}P^1$ over the origin? (Just asking, I know nothing about strictly psh functions, but the above is a basic example of a non-quasi-affine variety with no positive dimensional compact subvarieties.) $\endgroup$
    – Piotr Achinger
    Commented Jun 4, 2020 at 13:12

1 Answer 1

Reset to default
5
$\begingroup$

I don't think it's true. Take for instance $Y$ to be a non-Kahler compact complex surface which has no compact complex curves, for instance an Inoue surface. If $p\in Y$ is some arbitrary point, set $X=Y\setminus\{p\}$. Then obviously $X$ has no compact curve, and if $\varphi$ is some strictly plurisubharmonic function on $Y$, then $\omega:=i\partial\bar\partial\varphi$ is a positive $(1,1)$-current, which, since $p$ has codimension $2$ in $X$, can be extended to $Y$ by Shiffman's extension theorem. So you have $\tilde\omega$ a a closed positive current on $Y$ which is smooth and $>0$ on $X$, but then there is a (more or less) standard procedure which allows you to remove the singularity at $p$, and to obtain a smooth $(1,1)$-form, strictly positive on $Y$. This procedure is due to Miyaoka. But this is impossible since the surface $Y$ is non-Kahler.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .