Left-invariant Complex Structure Research Articles

CR embeddings of nilpotent Lie groups

In this note we show that a connected, simply connected nilpotent Lie group with an integrable left-invariant complex structure on a generating and suitably complemented subbundle of the tangent bundle admits a Cauchy-Riemann (CR) embedding in complex space defined by polynomials. We also show that a similar conclusion holds on suitable quotients of nilpotent Lie groups. Our results extend the CR embeddings constructed by Naruki [Publ. Res. Inst. Math. Sci. 6 (1970), pp. 113–187] in 1970. In particular, our generalisation to quotients allows us to see a class of Levi degenerate CR manifolds as quotients of nilpotent Lie groups.

K\"{a}hler-Norden structures on Hom-Lie groups and Hom-Lie algebras

The aim of this paper is to describe two geometric notions, holomorphic Norden structures and K\"{a}hler-Norden structures on Hom-Lie groups, and study their relationships in the left invariant setting. We study K\"{a}hler-Norden structures with abelian complex structures and give the curvature properties of holomorphic Norden structures on Hom-Lie groups. Finally, we show that any left-invariant holomorphic Hom-Lie group is a flat (holomorphic Norden Hom-Lie algebra carries a Hom-Left-symmetric algebra) if its left-invariant complex structure (complex structure) is abelian.

Levi-flat CR structures on compact Lie groups

Pittie (Proc Indian Acad Sci Math Sci 98:117-152, 1988) proved that the Dolbeault cohomology of all left-invariant complex structures on compact Lie groups can be computed by looking at the Dolbeault cohomology induced on a conveniently chosen maximal torus. We generalized Pittie’s result to left-invariant Levi-flat CR structures of maximal rank on compact Lie groups. The main tools we used was a version of the Leray–Hirsch theorem for CR principal bundles and the algebraic classification of left-invariant CR structures of maximal rank on compact Lie groups (Charbonnel and Khalgui in J Lie Theory 14:165-198, 2004) .

Plurisigned hermitian metrics

Let ( X , ω ) (X,\omega ) be a compact hermitian manifold of dimension n n . We study the asymptotic behavior of Monge-Ampère volumes ∫ X ( ω + d d c φ ) n \int _X (\omega +dd^c \varphi )^n , when ω + d d c φ \omega +dd^c \varphi varies in the set of hermitian forms that are d d c dd^c -cohomologous to ω \omega . We show that these Monge-Ampère volumes are uniformly bounded if ω \omega is “strongly pluripositive”, and that they are uniformly positive if ω \omega is “strongly plurinegative”. This motivates the study of the existence of such plurisigned hermitian metrics. We analyze several classes of examples (complex parallelisable manifolds, twistor spaces, Vaisman manifolds) admitting such metrics, showing that they cannot coexist. We take a close look at 6 6 -dimensional nilmanifolds which admit a left-invariant complex structure, showing that each of them admit a plurisigned hermitian metric, while only few of them admit a pluriclosed metric. We also study 6 6 -dimensional solvmanifolds with trivial canonical bundle.

Balanced Hermitian structures on almost abelian Lie algebras

We study balanced Hermitian structures on almost abelian Lie algebras, i.e. on Lie algebras with a codimension-one abelian ideal. In particular, we classify six-dimensional almost abelian Lie algebras which carry a balanced structure. It has been conjectured in [1] that a compact complex manifold admitting both a balanced metric and an SKT metric necessarily has a Kähler metric: we prove this conjecture for compact almost abelian solvmanifolds with left-invariant complex structures. Moreover, we investigate the behaviour of the flow of balanced metrics introduced in [2] and of the anomaly flow [3] on almost abelian Lie groups. In particular, we show that the anomaly flow preserves the balanced condition and that locally conformally Kähler metrics are fixed points.

Non-invariant deformations of left-invariant complex structures on compact Lie groups

Abstract We give small deformations of a left-invariant complex structure on each simply connected semisimple compact Lie group of even dimension which are not biholomorphic to any left-invariant (right-invariant) complex structure by using the Kuranishi space. On such deformed complex manifolds, we prove the Borel–Weil–Bott type theorem, and we compute the cohomology of holomorphic tangent bundles.

MAXIMAL NILPOTENT COMPLEX STRUCTURES

Let the pair (𝔤, J) be a nilpotent Lie algebra 𝔤 (NLA for short) endowed with a nilpotent complex structure J. In this paper, motivated by a question in the work of Cordero, Fernández, Gray and Ugarte [6], we prove that 2 ≤ v(J) ≤ 3 for (𝔤, J) when v(𝔤) = 2, where v(𝔤) is the step of 𝔤 and v(J) is the unique smallest integer such that 𝔞(J)v(J) = 𝔤 as in the [6, Def. 1, 8]. When v(𝔤) = 3, for arbitrary n ≥ 3, there exists a pair (𝔤, J) such that v(J) = dimℂ 𝔤 = n, for which we call the J in the pair (𝔤, J), satisfying v(J) = dimℂ 𝔤 = n, a maximal nilpotent (MaxN for short) complex structure. The algebraic dimension of a nilmanifold endowed with a left invariant MaxN complex structure is discussed. Furthermore, a structure theorem is proved for the pair (𝔤, J), where v(𝔤) = 3 and J is a MaxN complex structure.

Deformations of Dolbeault cohomology classes for Lie algebra with complex structures

In this paper, we study deformations of complex structures on Lie algebras and its associated deformations of Dolbeault cohomology classes. A complete deformation of complex structures is constructed in a way similar to the Kuranishi family. The extension isomorphism is shown to be valid in this case. As an application, we prove that given a family of left-invariant deformations $$\{M_t\}_{t\in B}$$ of a compact complex manifold $$M=(\Gamma \setminus G, J)$$ where G is a Lie group, $$\Gamma$$ a sublattice and J a left-invariant complex structure, the set of all $$t\in B$$ such that the Dolbeault cohomology on $$M_t$$ may be computed by left-invariant tensor fields is an analytic open subset of B.

Generalized Kähler almost abelian Lie groups

We study left-invariant generalized K\"ahler structures on almost abelian Lie groups, i.e., on solvable Lie groups with a codimension-one abelian normal subgroup. In particular, we classify six-dimensional almost abelian Lie groups which admit a left-invariant complex structure and establish which of those have a left-invariant Hermitian structure whose fundamental 2-form is $\partial \bar \partial$-closed. We obtain a classification of six-dimensional generalized K\"ahler almost abelian Lie groups and determine the 6-dimensional compact almost abelian solvmanifolds admitting an invariant generalized K\"ahler structure. Moreover, we prove some results in relation to the existence of holomorphic Poisson structures and to the pluriclosed flow.

Almost complex manifolds with small Nijenhuis tensor

Abstract We give several explicit examples of compact manifolds with a 1-parameter family of almost complex structures having arbitrarily small Nijenhuis tensor in the C 0-norm. The 4-dimensional examples possess no complex structure, whereas the 6-dimensional example does not possess a left invariant complex structure, and whether it possesses a complex structure appears to be unknown.

TWISTED DOLBEAULT COHOMOLOGY OF NILPOTENT LIE ALGEBRAS

It is well known that the cohomology of any non-trivial 1-dimensional local system on a nilmanifold vanishes (this result, due to J. Dixmier, was also announced and proved in some particular case by 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 Dixmier’s and Alaniya’s theorem, showing that the Dolbeault cohomology $$ {H}^{0,p}\left(\mathfrak{g},L\right) $$ 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, when the complex structure is rational). 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 Kahler 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.

On Bismut flat manifolds

In this paper, we give a classification of all compact Hermitian manifolds with flat Bismut connection. We show that the torsion tensor of such a manifold must be parallel, thus the universal cover of such a manifold is a Lie group equipped with a bi-invariant metric and a compatible left invariant complex structure. In particular, isosceles Hopf surfaces are the only Bismut flat compact non-Kähler surfaces, while central Calabi-Eckmann threefolds are the only simply-connected compact Bismut flat threefolds.

Complex nilmanifolds and Kähler-like connections

In this note, we analyze the question of when will a complex nilmanifold have Kähler-like Strominger (also known as Bismut), Chern, or Riemannian connection, in the sense that the curvature of the connection obeys all the symmetries of that of a Kähler metric. We give a classification in the second case and a partial description in the first and the third case. It would be interesting to understand these questions for all Lie–Hermitian manifolds, namely, Lie groups equipped with a left invariant complex structure and a compatible left invariant metric.

Lie groups with flat Gauduchon connections

We pursue the research line proposed in Yang and Zheng (Acta. Math. Sinica (English Series), 34(8):1259–1268, 2018) about the classification of Hermitian manifolds whose s-Gauduchon connection $$\nabla ^s =(1-\frac{s}{2})\nabla ^c + \frac{s}{2}\nabla ^b$$ is flat, where $$s\in {\mathbb {R}}$$ and $$\nabla ^c$$ and $$\nabla ^b$$ are the Chern and the Bismut connections, respectively. We focus on Lie groups equipped with a left invariant Hermitian structure. Such spaces provide an important class of Hermitian manifolds in various contexts and are often a valuable vehicle for testing new phenomena in complex and Hermitian geometry. More precisely, we consider a connected 2n-dimensional Lie group G equipped with a left-invariant complex structure J and a left-invariant compatible metric g and we assume that its connection $$\nabla ^s$$ is flat. Our main result states that if either n=2 or there exits a $$\nabla ^s$$ -parallel left invariant frame on G, then g must be Kähler. This result demonstrates rigidity properties of some complete Hermitian manifolds with $$\nabla ^s$$ -flat Hermitian metrics.

LEFT-INVARIANT ALMOST PARA-COMPLEX EINSTEINIAN STRUCTURES ON SIX-DIMENSIONAL NILPOTENT LIE GROUPS

As is well known, there are 34 classes of isomorphic simply connected six-dimensional nilpotent Lie groups. Of these, only 26 classes admit left-invariant symplectic structures and only 18 admit left-invariant complex structures. There are five six-dimensional nilpotent Lie groups G , which do not admit neither symplectic, nor complex structures and, therefore, can be neither almost pseudo- Kӓhlerian, nor almost Hermitian. In this work, these Lie groups are being studied. The aim of the paper is to define new left-invariant geometric structures on the Lie groups under consideration that compensate, in some sense, the absence of symplectic and complex structures. Weakening the closedness requirement of left-invariant 2-forms ω on the Lie groups, non-degenerated 2-forms ω are obtained, whose exterior differential dω is also non-degenerated in Hitchin sense [6]. Therefore, the Hitchin’s operator K dω is defined for the 3-form dω . It is shown that K dω defines an almost complex or almost para-complex structure for G and the couple ( ω, dω ) defines pseudo-Riemannian metrics of signature (2,4) or (3,3), which is Einsteinian for 4 out of 5 considered Lie groups. It gives new examples of multiparametric families of Einstein metrics of signature (3,3) and almost para-complex structures on six-dimensional nilmanifolds, whose structural group is being reduced to SL (3 , R) SO (3 , 3). On each of the Lie groups under consideration, compatible pairs of left-invariant forms (ω, Ω), where Ω = d ω, are obtained. For them the defining properties of half-flat structures are naturally fulfilled: d Ω = 0 and ωΩ = 0. Therefore, the obtained structures are not only almost Einsteinian para-complex, but also pseudo- Riemannian half-flat.

Some relations between Hodge numbers and invariant complex structures on compact nilmanifolds

Abstract Let N be a simply connected real nilpotent Lie group, n its Lie algebra, and € a lattice in N. If a left-invariant complex structure on N is Γ-rational, then HƏ̄ s,t(Γ/N) ≃ HƏ̄ s,t(nC) for each s; t. We can construct different left-invariant complex structures on one nilpotent Lie group by using the complexification and the scalar restriction. We investigate relationships to Hodge numbers of associated compact complex nilmanifolds.

Generalized deformations and holomorphic Poisson cohomology of solvmanifolds

We describe the generalized Kuranishi spaces of solvmanifolds with left-invariant complex structures. By using such description, we study the stability of left-invariantness of deformed generalized complex structures and smoothness of generalized Kuranishi spaces on certain classes of solvmanifolds. We also give explicit finite-dimensional cochain complexes which computes the holomorphic Poisson cohomology of nilmanifolds and solvmanifolds.

Complex, Symplectic, and Contact Structures on Low-Dimensional Lie Groups

It is well known that on any Lie group, a left-invariant Riemannian structure can be defined. For other left-invariant geometric structures, for example, complex, symplectic, or contact structures, there are difficult obstructions for their existence, which have still not been overcome, although a lot of works were devoted to them. In recent years, substantial progress in this direction has been made; in particular, classification theorems for low-dimensional groups have been obtained. This paper is a brief review of left-invariant complex, symplectic, pseudo-Kahlerian, and contact structures on low-dimensional Lie groups and classification results for Lie groups of dimension 4, 5, and 6.

The Frölicher Spectral Sequence of Certain Solvmanifolds

We show that the Frolicher spectral sequence of a complex parallelizable solvmanifold is degenerate at the E 2-term. For a semi-direct product $G=\mathbb{C}^{n}\ltimes_{\phi}N$ of Lie groups with lattice Γ=Γ′⋉Γ′′ such that N is a nilpotent Lie group with a left-invariant complex structure and ϕ is a semi-simple action, we also show that, if the Frolicher spectral sequence of the nilmanifold N/Γ′′ is degenerate at the E r -term for r≥2, then the Frolicher spectral sequence of the solvmanifold G/Γ is also degenerate at the E r -term.

On balanced Hermitian structures on Lie groups

We present several methods for the construction of balanced Hermitian structures on Lie groups. In our methods a partial differential equation is involved so that the resulting structures are in general non homogeneous. In particular, we prove that for 3-step nilpotent Lie groups G of dimension 6, any left-invariant complex structure on G admits a balanced Hermitian metric. Starting from normal almost contact structures, we construct balanced metrics on 6-dimensional manifolds, generalizing warped products. Finally, explicit balanced Hermitian structures are also given on solvable Lie groups defined as semidirect products \({\mathbb{R}^k \ltimes \mathbb{R}^{2n-k}}\).