Iwasawa manifold

Example

Let $G$ be the complex Heisenberg group of matrices $$\begin{pmatrix} 1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1 \end{pmatrix}, \qquad x, y, z \in \C,$$ a complex Lie group isomorphic to $\C^3$ as a complex manifold. Let $\Gamma \subset G$ be the subgroup of matrices with $x, y, z$ in the Gaussian integers $\Z[i]$. The Iwasawa manifold is the quotient $X = \Gamma \backslash G$ of $G$ by the left action of $\Gamma$. It is a compact complex manifold of dimension $3$: $\Gamma$ is a discrete cocompact subgroup, a lattice of $G$, and acts freely by holomorphic maps.

Check that $X$ is not Kähler. Left multiplication by the matrix with entries $(a, b, c)$ sends $(x, y, z)$ to $(x + a, \, y + b, \, z + a y + c)$. So the holomorphic $1$-forms $dx$, $dy$ and $\theta = dz - x\, dy$ are invariant: $d(z + ay + c) - (x + a)\, d(y + b) = dz - x\, dy$. They descend to holomorphic $1$-forms on $X$. But $d\theta = -dx \wedge dy \ne 0$, so $\theta$ is not closed. On a compact Kähler manifold every holomorphic form is closed, so $X$ is not Kähler.