I am looking for examples of non-Kähler compact complex manifolds of dimension four with trivial canonical class and $H^i(M, {\mathcal O}_M) = H^0(M, \Omega^i_M) =0$ for $0< i < \dim M$.

In dimension two, no such manifolds exist (compact complex surfaces with trivial canonical class and $H^1(M, {\mathcal O}_M)=0$ are K3 surfaces and they are all Kähler.)

In dimension three, Infinitely many topological types have been constructed by R. Friedman.

My question is: "Are there any examples of such manifolds in dimension four?"

It is notable that Guan constructed examples of non-Kähler compact complex symplectic fourfold with trivial canonical class.


1 Answer 1


Examples were recently constructed by Lee and Sano by two different constructions (which both involve smoothing varieties with normal crossings).

Sano's paper also constructs examples in all dimensions $\geq 4$ with arbitrarily large second Betti number.


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