Example 11.47.151E30DC99E57B

Proof. Take $Y=(\mathbb C^2\setminus \{ 0\} )/\langle \zeta \mapsto \mathrm e\zeta \rangle $. With $s=\log |\zeta |^2$, the form $h_0=\mathrm{dd}^{\mathrm{c}}s+\mathrm ds\wedge \mathrm d^{\mathrm c}s$ is invariant under the quotient. Its two summands are positive on the tangential and radial complex directions, respectively, so it is Hermitian. Moreover $\mathrm dh_0=-\mathrm ds\wedge \mathrm{dd}^{\mathrm{c}}s\ne 0$. The Hopf surface is diffeomorphic to $S^3\times S^1$ and has odd first Betti number. Thus $X=\mathbb P^2\times Y$ is a non-Kähler fourfold with $\mathrm dH\ne 0$, and proposition 11.47.14 applies. $\square$