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I am interested in a simply-connected compact complex manifold $M$ of dimension three with trivial canonical class. Note that if it is K"ahler, then it is a Calabi-Yau threefold. Its independent Betti numbers are $b_2, b_3$ and $b_3$ is a positive number.

I am interested in cases where $b_2, b_3$ are particularly small.

When $b_2=0$, Examples with $b_3 = 4, 6, 8 ...$ are constructed by Lu and Tian.

When $b_2 =1$ (which is the next simplest case), the classification seems wide open. I am wondering if an example with $b_3=2$ exists in this case.

Let me state my question precisely:

Is there any known example with $b_2=1, b_3=2$? If not, is there any possible reason that such an example does not exist?

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    $\begingroup$ "It is known that an example with $b_3=2$ does not exist" : could you give a reference for that? And why does Serre duality imply $b_3>0$? $\endgroup$
    – abx
    Commented Jan 13, 2023 at 6:45
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    $\begingroup$ The paper arxiv.org/pdf/1506.00892.pdf constructs a projective Calabi-Yau threefold with $b_2=b_3=2$. But it is not simply connected. At least in the projective case, the examples with small Hodge numbers I am aware of almost always use a free action on a larger CY3 to cut down the number of divisors. See for example arxiv.org/pdf/1602.06303.pdf. $\endgroup$
    – Balazs
    Commented Jan 13, 2023 at 12:49
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    $\begingroup$ @abx: if $X$ is a compact complex manifold whose canonical bundle has a non-zero holomorphic section then the real and imaginary parts are closed and give two independent elements in middle degree de Rham cohomology. In particular, in the case the OP is interested in, $b_3 \geq 2$. $\endgroup$
    – Joel Fine
    Commented Jan 13, 2023 at 15:42
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    $\begingroup$ In a paper I wrote with Dima Panov, you can find infinitely many distinct complex structures with trivial canonical bundle on the manifold $2(S^3\times S^3)\#(S^2 \times S^4)$. This has $b_2=1$ and $b_3 = 4$. See arxiv.org/abs/0905.3237. $\endgroup$
    – Joel Fine
    Commented Jan 13, 2023 at 15:52
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    $\begingroup$ How do you conclude that for any complex structure on $S^3 \times S^3$ the canonical bundle is non-trivial? $\endgroup$
    – Aleksandar Milivojević
    Commented Jan 13, 2023 at 20:53

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