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Yesterday I asked the following question to which abx has given a positive answer.

examples of Kähler manifolds with trivial odd Betti numbers and first Chern classes

But I suddenly realized that those Fano manifolds I mentioned in the quesion above (flag manifolds $G/P$ and Fano contact manifolds) not only have trivial odd Betti numbers but have stronger properties: their Hodge numbers satisfy $h^{p,q}=0$ whenever $p\neq q$.

But the example abx mentioned (some smooth complete intersections) and the K3 surfaces seem not to support the above property.

So I would like to modify my original question by asking the following: does there exist a compact Kähler manifold whose first Chern class is trivial such that its Hodge numbers satisfy $h^{p,q}=0$ whenever $p\neq q$?

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    $\begingroup$ It depends on what do you mean by vanishing of the first Chern class. On one hand, Enriques surfaces have $c_1 = 0$ in cohomology with $\mathbb{Q}$ coefficients and its Hodge diamond is diagonal. On the other hand, if the canonical class of $X$ is trivial then $h^{n,0}(X) \ne 0$. $\endgroup$
    – Sasha
    Commented May 25, 2014 at 9:01
  • $\begingroup$ @Sasha What I mean is $c_1=0$ in $H^{2}(M;\mathbb{R})$. I want to know if, for any $n$, there exists an $n$-dimensional example. $\endgroup$
    – Kevin
    Commented May 25, 2014 at 11:37
  • $\begingroup$ @Kevin: In that case Sasha's examples are valid -- you are killing the torsion in cohomology when you use $\mathbb{R}$-coefficients. So there are Enriques surfaces, and many more examples in dimensions $>2$. $\endgroup$
    – Jason Starr
    Commented May 25, 2014 at 13:11

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If the Chern class vanishes over integers, it's Calabi-Yau manifold, and it has a holomorphic (n,0)-form by Bogomolov's theorem (see there: Two definitions of Calabi-Yau manifolds). If you relax your condition by asking $c_1$ to vanish over reals, then there are many examples, such as an Enriques surface; its only non-zero Hodge numbers are $h^{1,1}=10$, $h^{0,0}=1$, $h^{2,2}=1$.

The quotients of Calabi-Yau manifolds often satisfy $h^{p,q}=0$ for all $p\neq q$. However, there is an interesting question (still unresolved, as far as I know): does there exist a quotient of a compact torus which is Kahler and has this property?

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