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Twisted Dolbeault cohomology of nilpotent Lie algebras

Verbitsky M., Ornea L.
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 Kähler structure on its cover M̃ such that the deck transform map acts on M̃ 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.