We obtain several restrictions on the terms of the ascending central series of a nilpotent Lie algebra g under the presence of a complex structure J. In particular, we find a bound for the dimension of the center of g when it does not contain any non-trivial J-invariant ideal. Thanks to these results, we provide a structural theorem describing the ascending central series of 8-dimensional nilpotent Lie algebras g admitting this particular type of complex structures J. Since our method is constructive, it allows us to describe the complex structure equations that parametrize all such pairs pg, Jq.

The Bott-Chern cohomology of six-dimensional nilmanifolds endowed with invariant complex structure is studied with special attention to the cases when balanced or strongly Gauduchon Hermitian metrics exist. We consider complex invariants introduced by Angella and Tomassini and by Schweitzer, which are related to the ∂∂-lemma condition and defined in terms of the Bott-Chern cohomology, and show that the vanishing of some of these invariants is not a closed property under holomorphic deformations. In the balanced case, we determine the spaces that parametrize deformations in type IIB supergravity described by Tseng and Yau in terms of the Bott-Chern cohomology group of bidegree (2, 2). Int. J. Math. 2014.25. Downloaded from www.worldscientific.com by THE UNIVERSITY OF OKLAHOMA on 02/04/15. For personal use only.On the Bott-Chern cohomology and balanced Hermitian nilmanifolds fluxes in the presence of O5/D5-brane sources. More precisely, the moduli space of solutions given by linearized variations is parametrized by the space, which we will denote by L 2,2 (M, J, F ), consisting of the harmonic forms of the Bott-Chern cohomology group H 2,2 BC (M, J) which are annihilated by F . Using the explicit description of H 2,2 BC (M ) given in Sec. 3 and the results of [35], we determine the space L 2,2 (Γ\G, J, F ) for any invariant balanced Hermitian structure (J, F ) on a 6-dimensional nilmanifold M = Γ\G. We prove that its dimension only depends on the complex structure; in particular, if M 0 denotes the Iwasawa manifold then one has that dim L 2,2 (M 0 , F ) = 7 for any F . As an application we show that dim L 2,2 is not stable under small deformations; concretely, there is a holomorphic deformation M t , t ∈ C with |t| < 1, of the Iwasawa manifold M 0 admitting balanced structures for each t and such that dim L 2,2 (M t , F ) = 5 for 0 < |t| < 1 and for any balanced Hermitian metric F on M t (Proposition 5.4).During the preparation of this paper we were informed by Adriano Tomassini that Angella, Franzini and Rossi have obtained in [5] similar computations which are used to provide a measure of the degree of non-Kählerianity of 6-dimensional nilmanifolds with invariant complex structure, as well as the relation between Bott-Chern cohomological properties and existence of pluriclosed metrics. Invariant Complex Structures on Nilmanifolds, Bott-Chern and Aeppli Cohomologies, and Special Hermitian Metrics1450057-3 Int. J. Math. 2014.25. Downloaded from www.worldscientific.com by THE UNIVERSITY OF OKLAHOMA on 02/04/15. For personal use only. (M ) = h 0,1 BC (M ). Corollary 2.1. Let M be a 2n-dimensional nilmanifold (not a torus) endowed with an abelian complex structure J. Then, the map (2.4) is not injective. Proof. It suffices to show that if J is abelian then h 0,1 (M ) > h 0,1 BC (M ). By Theorem 2.1 we have H 0,1 ∂ (M ) ∼ = H 0,1 ∂ (g) = {α 0,1 ∈ g 0,1 |∂α 0,1 = 0} ∼ = {α 1,0 ∈ g 1,0 | ∂α 1,0 = 0}, 1450057-5 Int. J. Math. 2014.25. Downloaded from www.worldscientific.com by THE UNIVERSITY OF OKLAHOMA on 02/04/15. For pe...

Abstract. In this note we construct, for every n ≥ 4, a non-Kähler compact complex manifold X of complex dimension n admitting a balanced metric and an astheno-Kähler metric which is in addition k-th Gauduchon for any 1 ≤ k ≤ n − 1.

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We find a one-parameter family of non-isomorphic nilpotent Lie algebras ga, with a ∈ [0, ∞), of real dimension eight with (strongly non-nilpotent) complex structures. By restricting a to take rational values, we arrive at the existence of infinitely many real homotopy types of 8dimensional nilmanifolds admitting a complex structure. Moreover, balanced Hermitian metrics and generalized Gauduchon metrics on such nilmanifolds are constructed.2000 Mathematics Subject Classification. Primary 55P62, 17B30; Secondary 53C55.