Are there examples of compact complex manifolds $X$ with $K\_X$ nef, but $X$ is not Kähler? Perhaps even non-Moishezon examples?


Here, nef can be defined as follows: For any $\varepsilon>0$ there is a Hermitian metric $h\_{\varepsilon}$ on $K\_X$ with curvature $\Theta\_{h\_{\varepsilon}} \geq - \varepsilon \omega$, where $\omega$ is a positive-definite real $(1,1)$--form on $X$.



Compact complex non-Kähler manifolds with nef canonical bundle
Let $X$ and $Y$ be compact complex manifolds. Note that $K\_{X\times Y} \cong \pi\_1^\*K\_X\otimes \pi\_2^\*K\_Y$. If $Y$ has trivial canonical bundle, then $K\_{X\times Y} \cong \pi\_1^\*K\_X$. Now the pullback of a nef line bundle is again nef, see Proposition 1.8 (i) of [*Compact Complex Manifolds with Numerically Effective Tangent Bundles*](https://www-fourier.ujf-grenoble.fr/%7Edemailly/manuscripts/dps1.pdf) by Demailly, Peternell, and Schneider. So one can construct many examples by choosing $X$ with $K\_X$ nef and $Y$ non-Kähler with $K\_Y$ trivial.


**Example:** Let $X$ be a curve of genus $g > 1$ and $Y$ be a primary Kodaira surface. Then $X\times Y$ is a non-Kähler threefold with $K\_{X\times Y}$ nef. Note that $X\times Y$ is also not Moishezon as it contains $Y$ as a complex submanifold and $Y$ is not Moishezon.


For more examples of non-Kähler manifolds with $K\_Y$ trivial, see the introduction of [*Non-Kähler Calabi-Yau Manifolds*](https://arxiv.org/abs/1401.4797) by Tosatti. As is pointed out in Proposition 1.1, if $K\_Y$ is trivial (or even just torsion), then it admits a metric $h$ with curvature $\Theta\_h = 0$ and hence $K\_Y$ is nef, so we don't even need to take a product with $X$ in the above construction.


