# Twisted Dolbeault cohomology of nilpotent Lie algebras

## TL;DR

This paper proves a Dolbeault cohomology vanishing theorem for nilpotent Lie algebras with non-trivial local system coefficients, revealing limitations of existing theorems and characterizing LCK nilmanifolds.

## Contribution

It establishes a Dolbeault cohomology vanishing result for nilpotent Lie algebras with non-trivial local systems, and characterizes LCK structures on nilmanifolds as Vaisman.

## Key findings

- Dolbeault cohomology of non-trivial local systems on nilpotent Lie algebras vanishes.
- The twisted version of Console-Fino theorem is false.
- Any LCK nilmanifold is Vaisman and is a quotient of a Heisenberg group product.

## Abstract

It is well known that cohomology of any non-trivial 1-dimensional local system on a nilmanifold vanishes (this result is due to L. 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 Alaniya's theorem, showing that the Dolbeault cohomology 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). 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 K\"ahler 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.