Define a compact Kähler surface $X$ to be a K3 surface if $X$ is simply connected, $K\_X \simeq \mathcal{O}\_X$, and $h^{0,1}=0$. If $X$ is projective, then a theorem typically attributed to Bogomolov and Mumford, asserts that $X$ admits a rational curve. It has recently been shown that projective K3 surfaces admit infinitely many rational curves.



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> 1. Do non-projective K3 surfaces admit rational curves?
> 2. Do non-projective K3 surfaces admit infinitely many rational curves?
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Do non-projective K3 surfaces have rational curves?
Some of them do, and some don't.


Indeed, by global Torelli theorem, there is a K3 surface $X$ with $\mathrm{Pic}(X) = 0$. Such $X$ has no curves, in particular no rational curves.


On the other hand, there is a K3 surface $X$ such that $\mathrm{Pic}(X)$ is generated by a single class with square $-2$; such a class (up to sign) is represented by a smooth rational curve.


