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By a threefold, I mean a compact complex manifold of dimension three.

For a simply-connected threefold with trivial canonical class, its Betti numbers satisfy: $$b_2 \ge 0, b_3 \ge 2.$$

I am wondering whether there is such a threefold with those minimal Betti numbers. Precisely,

Is there a simply-connected threefold with trivial canonical class and Betti numbers $b_2 = 0$, $b_3 = 2$?

Note that such a threefold cannot be projective.

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    $\begingroup$ How about the Hopf manifold $S^3 \times S^3$? $\endgroup$
    – Nick L
    Commented Jan 11, 2023 at 8:17
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    $\begingroup$ @Nick L: Hopf manifolds are diffeomorphic to $S^{1}\times S^{2n-1}$. I guess you are thinking of the Calabi-Eckmann manifolds, but their canonical class is not trivial. $\endgroup$
    – abx
    Commented Jan 11, 2023 at 9:07
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    $\begingroup$ Topologically any simply connected threefold with $b_2= 0$ is a connected sum $k(S^3 \times S^3) \# Q$ where $Q$ is a rational homology sphere. Lu and Tian showed in "The complex structure on a connected sum of $S^3\times S^3$ with trivial canonical bundle" that $25(S^3 \times S^3)$ carries a complex structure with trivial canonical. I don't know if smaller examples have been found. $\endgroup$
    – Aleksandar Milivojević
    Commented Jan 11, 2023 at 9:14
  • $\begingroup$ @Aleksandar Milivojević: Could you give a reference for your statement "Topologically any .... homology sphere"? $\endgroup$
    – Basics
    Commented Jan 11, 2023 at 17:49
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    $\begingroup$ In their subsequent paper "The complex structures on connected sums of $S^3\times S^3$" Manifolds and geometry (Pisa, 1993), 284–293, Sympos. Math., XXXVI, Cambridge Univ. Press, Cambridge, 1996, Lu and Tian construct complex structures with trivial canonical bundle on all $k(S^3\times S^3)$ for $k\geq 2$. $\endgroup$
    – YangMills
    Commented Jan 11, 2023 at 18:27

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