TWISTED DOLBEAULT COHOMOLOGY OF NILPOTENT LIE ALGEBRAS

Abstract

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.