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Assume that $M$ is a closed connected Ricci-flat Kaehler manifold $M$ of complex dimension $n\geq 3$ with $h^{2,0}(M)=0$. Is is possible that

  • $h^{n, 0}(M)\neq 1$
  • $h^{p, 0}(M)\neq 0$ for some $0< p< n$?
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    $\begingroup$ What about a product of $r$ factors, each of which is a Calabi-Yau threefold? By Kuenneth, $h^{3,0}(M)$ is $r$-dimensional. $\endgroup$ – Jason Starr Dec 10 '18 at 18:18
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After replacing by a finite covering space, you must have $h^{n0}=1$, because you have a parallel volume form, so nowhere zero, and any other holomorphic volume form is a holomorphic multiple, so a constant multiple (as holomorphic functions are constant).

Kobayashi, First Chern class and holomorphic tensor fields, 1980, proved that any holomorphic tensor on a compact Ricci-flat Kaehler manifold is parallel. Hence it induces a reduction of holonomy group. The holonomy group is a product of groups arising as irreducible holonomy groups, i.e. copies of the special unitary group $SU(n_i)$ and the compact symplectic group $Sp(n_j)$. But since you have $h^{20}=0$, you have no $Sp(n_j)$ factors, so a product of $SU(n_i)$ factors, i.e. locally a product of Calabi-Yau and flat Kaeher metrics. Unless there is a flat factor in the product, you don't get any parallel tensors except for the holomorphic $n_j$-forms. By a theorem of Beauville (J.D.G. 1983), after a finite covering, you have a product of simply connected Calabi Yaus with irreducible $SU(n_i)$ holonomy, $n_i\ge 3$, and a flat complex torus. But to have $h^{20}=0$, that torus is an elliptic curve.

As Martin de Borbon points out, there are examples of closed complex manifolds with Ricci flat Kaehler metrics with torsion canonical bundle, and then you will get $h^{n0}=0$, but after taking a finite covering space you get $h^{n0}=1$.

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  • $\begingroup$ if I understand correctly, your first paragraph answers the first question (we have to have $h^{n, 0}=1$). But your second paragraph I don't quite understand. We require from the outset that $h^{2, 0}=0$ and ask if some other $h^{p, 0}$ can be non-zero (excluding $p=0, n$). $\endgroup$ – geometer Dec 10 '18 at 17:21
  • $\begingroup$ Ok, I didn't read carefully enough. I will edit. $\endgroup$ – Ben McKay Dec 10 '18 at 17:30
  • $\begingroup$ Closed Ricci flat does not imply trivial canonical bundle. A product of an Enriques surface with an elliptic curve has no l holomorphic volume from $\endgroup$ – Martin de Borbon Dec 10 '18 at 21:57
  • $\begingroup$ @MartindeBorbon: ok, up to finite covering then. I will edit the answer again. $\endgroup$ – Ben McKay Dec 11 '18 at 5:45

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