Compact nilmanifolds with nilpotent complex structures: Dolbeault cohomology

Abstract

Abstract. We consider a special class of compact complex nilmanifolds, which we call compact nilmanifolds with nilpotent complex structure. It is shown that if Γ\G is a compact nilmanifold with nilpotent complex structure, then the Dolbeault cohomology H * , * ∂ (Γ\G) is canonically isomorphic to thē ∂-cohomology H * , * ∂ (g C ) of the bigraded complex (Λ * , * (g C ) * ,∂) of complex valued left invariant differential forms on the nilpotent Lie group G.

Families of strong KT structures in six dimensions

“…Using de 5 , de 6 in place of σ 1 , σ 2 is merely a change of real basis and must yield a matrix congruent to B. It follows that, in the above examples, B is the zero matrix for (i) and (v), B has rank 1 for (iii) and (vi), det B = 0 for (ii) and (iv).…”

“…Setting 6 gives dα 3 = α 12 , so J is integrable and abelian. Observe thatĴ has negative orientation, so its connected component in C does not contain ±J 0 .…”

Stability of Abelian Complex Structures

“…a compact quotient of a simply-connected nilpotent Lie group G by a uniform discrete subgroup Γ. We assume that M has an invariant complex structure J, that is to say that J comes from a complex structure J on the Lie algebra g of G. An important class of complex structures is given by the abelian ones [1,2], which are particular types of nilpotent complex structures considered in [4].…”

“…where g * (p,q) denotes the space of (p, q)-forms on g. Now assume that the Lie algebra g is k-step nilpotent and set n := dim C g. Let us use like in [4] the ascending series {g ℓ } with…”

“…, X n ,X n } is a real basis of g ℓ , n ℓ = dim C g ℓ . In fact, proceeding as in the proof of Theorem 12 in [4], having chosen a basis…”