Vaisman metrics on solvmanifolds and Oeljeklaus-Toma manifolds

Abstract

We prove the non‐existence of Vaisman metrics on some solvmanifolds with left‐invariant complex structures. By this theorem, we show that Oeljeklaus–Toma manifolds do not admit Vaisman metrics.

Hodge decomposition for Cousin groups and for Oeljeklaus-Toma manifolds

“…By a straightforward computation that results from (4.10) being an equality, we also have the following: Remark 4.9. In [Kas13] it is shown that any OT manifold X admits a solvmanifold presentation Γ \ G, in such a way that the natural complex structure on G is G-left invariant. It is well known that the Lie algebra cohomology H * (g) injects into H * dR (X).…”

Locally conformally Kähler solvmanifolds: a survey

“…• Kasuya proved in [56] that OT manifolds do not admit any Vaisman metric, by studying the Morse-Novikov cohomology H * θ of solvmanifolds Γ\G equipped with LCS structures, where θ denotes the corresponding Lee form of the LCS structure. • Ornea and Verbitsky proved in [76] that OT manifolds of type (s, 1) contain no nontrivial complex submanifolds.…”

Example of a six-dimensional LCK solvmanifold

“…[14]). Kasuya [7] showed that OT-manifolds are in fact solvmanifolds, and proved that OT-manifold of type (s, 1) has no Vaisman structures. We see that the solvmanifold Γ∖G is OT-manifold of type (2,1).…”

“…[7] Let g be the solvable Lie algebra of G. Then g * has a basis {λ i , µ i , υ,ῡ} such that [15] proved that the first Betti number of a Vaisman manifold is odd. Thus, the solvmanifold Γ∖G has a LCK structure, but it has no Vaisman structures.…”