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I am looking for interesting examples of smooth, projective threefolds $X$ (preferably over a number field) such that $ H^0(X, \Omega^2_X) $ or equivalently $H^2(X, \mathcal{O}_X)$ has dimension one. Who can provide such examples?

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  • $\begingroup$ By "interesting" do you mean threefolds which are not $\mathbb{P}^{1}$ bundles over a surface with $h^{2,0}=1$? $\endgroup$
    – Yusuf Mustopa
    Commented Jun 27, 2013 at 15:21
  • $\begingroup$ Not necessarily $\endgroup$
    – 3fold
    Commented Jun 27, 2013 at 15:49

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You can prove that such threefolds are either P^1-fibration over a base, which is a surface with $h^{2,0}=1$, or have pseudoeffective canonical bundle: arXiv:1304.7891, Corollary 4.3. In the later case you can run the minimal model program, obtaining that your variety is either general type or is a (singular) fibration with Calabi-Yau fibers over a general type variety with canonical singularities. However, since $h^{2,0}=1$, your base has $h^{2,0}<2$, and this restriction is pretty strong: either you have an elliptic fibration over a surface, or a K3 or toric fibration over an elliptic curve.

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    $\begingroup$ One comment: a $\mathbb{P}^1$-fibration in this context may only be etale locally trivial (away from the discriminant locus). There are "conic bundles" that are not Zariski locally trivial over surfaces with $h^{2,0} =1$. $\endgroup$
    – Jason Starr
    Commented Jun 28, 2013 at 17:25
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Let $S$ be a K3 surface, and let $S^{[2]}$ be the Hilbert scheme parametrizing length-2 subschemes of $S.$ Then $S^{[2]}$ is a smooth projective 4-fold with $h^{2,0}=1.$ If $D \subset S^{[2]}$ is a smooth ample divisor on $S^{[2]},$ then Kodaira vanishing implies that $D$ is a smooth projective threefold with $h^{2,0}=1.$

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