It is well known that the cohomology of any non-trivial 1-dimensional local system on a nilmanifold vanishes (this result, due to J. Dixmier, was also announced and proved in some particular case by Alaniya). A complex nilmanifold is a quotient of a nilpotent Lie group equipped with a left-invariant complex structure by an action of a discrete, co-compact subgroup. We prove a Dolbeault version of Dixmier’s and Alaniya’s theorem, showing that the Dolbeault cohomology \( {H}^{0,p}\left(\mathfrak{g},L\right) \) of a nilpotent Lie algebra with coefficients in any non-trivial 1-dimensional local system vanishes. Note that the Dolbeault cohomology of the corresponding local system on the manifold is not necessarily zero. This implies that the twisted version of Console–Fino theorem is false (Console–Fino proved that the Dolbeault cohomology of a complex nilmanifold is equal to the Dolbeault cohomology of its Lie algebra, when the complex structure is rational). As an application, we give a new proof of a theorem due to H. Sawai, who obtained an explicit description of LCK nilmanifolds. An LCK structure on a manifold M is a Kähler structure on its cover \( \tilde{M} \) such that the deck transform map acts on \( \tilde{M} \) by homotheties. We show that any complex nilmanifold admitting an LCK structure is Vaisman, and is obtained as a compact quotient of the product of a Heisenberg group and the real line.

众所周知，尼尔迈尼法尔上任何非平凡的一维局部系统的同调性都消失了（由于J. Dixmier，这一结果在Alaniya的某些特殊情况下也已宣布并证明）。复杂的nilmanifold是通过离散的共紧实亚组的作用而配备有左不变复杂结构的幂等Lie组的商。我们证明了Dixmier定理和Alaniya定理的Dolbeault版本，表明Dolbeault同调\（{H} ^ {0，p} \ left（\ mathfrak {g}，L \ right）\）在任何非平凡的一维局部系统中具有系数的幂等李代数的消失。请注意，流形上相应局部系统的Dolbeault同调不一定是零。这意味着Console-Fino定理的扭曲形式是错误的（Console-Fino证明了当复杂结构是有理数时，复杂的nilmanifold的Dolbeault同调等于其Lie代数的Dolbeault同调）。作为应用，由于H. Sawai给出了一个定理的新证明，H。Sawai获得了LCK nilmanifolds的明确描述。流形M上的LCK结构是其封面\（\ tilde {M} \）上的Kähler结构，使得甲板变换图作用于\（\ tilde {M} \）通过同质性。我们表明，任何接受LCK结构的复杂尼尔曼折叠都是Vaisman，并且作为Heisenberg群和实线乘积的紧凑商获得。