Non-Kählerian Compact Complex Surfaces

Parallel and totally umbilical hypersurfaces of the four‐dimensional Thurston geometry Sol04$\text{Sol}^4_0$

“…However, both of them are globally conformal Kähler, meaning that g is globally conformal to a Kähler metric, namely e 2𝑡 g. This can be checked by verifying that the two-forms e 2𝑡 Ω + and e 2𝑡 Ω − are closed, where Ω ± (⋅ , ⋅) = g(⋅ , 𝐽 ± ⋅). It is also worth mentioning that, as a complex manifold, Sol 4 0 is holomorphically isometric to the universal covering of an Inoue surface [12] equipped with Tricerri metric [18,20].…”

“…This can be checked by verifying that the two‐forms $$\text{e}^{2t} \Omega _+$$ and $$\text{e}^{2t} \Omega _-$$ are closed, where $$\Omega _\pm (\cdot \,,\cdot)=\tilde{g}(\cdot \,,J_\pm \,\cdot)$$. It is also worth mentioning that, as a complex manifold, $$\mathrm{Sol}_{0}^{4}$$ is holomorphically isometric to the universal covering of an Inoue surface [12] equipped with Tricerri metric [18, 20].…”