Console–Fino conjecture on invariant Dolbeault cohomology of nilmanifolds

Let $N=\left.\Gamma\right\backslash G$ be a nilmanifold endowed with a $G$-left-invariant complex structure $J$, and let $\mathfrak{g}$ be the Lie algebra naturally associated to $G$. The inclusion of the complex of $G$-left-invariant forms into the Dolbeault complex of $N$ induces a map

$$i\colon H^{\bullet,\bullet}_{\overline{\partial}}\left(\mathfrak{g}_{\mathbb{C}}\right)\longrightarrow H^{\bullet,\bullet}_{\overline{\partial}}(N).$$

Console–Fino conjecture. These maps of complexes are quasi-isomorphisms, equivalently,

$$i\colon H^{\bullet,\bullet}_{\overline{\partial}}\left(\mathfrak{g}_{\mathbb{C}}\right)\stackrel{\simeq}{\longrightarrow}H^{\bullet,\bullet}_{\overline{\partial}}(N).$$

The conjecture asserts that Dolbeault cohomology on every nilmanifold with a left-invariant complex structure is computed by left-invariant forms. The cited results establish that the property is open in the space of left-invariant complex structures, but do not establish it for all such structures.