Let $R$ be a commutative ring with identity. The *three-dimensional Heisenberg group* over $R$ is

$$\mathbb{H}(3, R) = \left\{\begin{bmatrix} 1 & z^1 & z^3\\ 0 & 1 & z^2\\ 0 & 0 & 1\end{bmatrix} : z^1, z^2, z^3 \in R\right\}.$$

The *Iwasawa manifold* $\mathbb{I}_3$ is the quotient of $\mathbb{H}(3, \mathbb{C})$ by the discrete subgroup $\mathbb{H}(3, \mathbb{Z}[i])$ acting on the left, i.e. $\mathbb{I}_3 := \mathbb{H}(3, \mathbb{Z}[i])\setminus\mathbb{H}(3, \mathbb{C})$. It is a compact complex three-fold with $h^{1,0}(\mathbb{I}_3) = 3$ and $h^{0,1}(\mathbb{I}_3) = 2$; see section 3.2.1.1 of *Cohomological Aspects in Complex Non-Kähler Geometry* by Angella. The holomorphic one-forms $dz^1$, $dz^2$, and $dz^3 - z^1dz^2$ on $\mathbb{H}(3, \mathbb{C})$ descend $\mathbb{I}_3$ and form a basis for $H^{1,0}_{\bar{\partial}}(\mathbb{I}_3)$.