We complete the classification of six-dimensional strongly unimodular almost nilpotent Lie algebras admitting complex structures. For several cases we describe the space of complex structures up to isomorphism. As a consequence we determine the six-dimensional almost nilpotent solvmanifolds admitting an invariant complex structure and study the existence of special types of Hermitian metrics, including SKT, balanced, locally conformally Kähler, and strongly Gauduchon metrics. In particular, we determine new balanced solvmanifolds and confirm a conjecture by the first author and Vezzoni regarding SKT and balanced structures in the six-dimensional strongly unimodular almost nilpotent case. Moreover, we prove some negative results regarding complex structures tamed by symplectic forms, showing in particular that in every dimension such structures cannot exist on non-Kähler almost abelian Lie algebras.
An Einstein nilradical is a nilpotent Lie algebra, which can be the nilradical of a metric Einstein solvable Lie algebra. The classification of Riemannian Einstein solvmanifolds (possibly, of all noncompact homogeneous Einstein spaces) can be reduced to determining, which nilpotent Lie algebras are Einstein nilradicals and to finding, for every Einstein nilradical, its Einstein metric solvable extension. For every nilpotent Lie algebra, we construct an (essentially unique) derivation, the pre-Einstein derivation, the solvable extension by which may carry an Einstein inner product. Using the pr…
A Riemannian Einstein solvmanifold (possibly, any noncompact homogeneous Einstein space) is almost completely determined by the nilradical of its Lie algebra. A nilpotent Lie algebra, which can serve as the nilradical of an Einstein metric solvable Lie algebra, is called an Einstein nilradical. Despite a substantial progress towards the understanding of Einstein nilradicals, there is still a lack of classification results even for some well-studied classes of nilpotent Lie algebras, such as the two-step ones. In this paper, we give a classification of two-step nilpotent Einstein nilradicals in…
In the homogeneous case, the only curvature behavior which is still far from being understood is Ricci negative. In this paper, we study which nilpotent Lie algebras admit a Ricci negative solvable extension. Different unexpected behaviors were found. On the other hand, given a nilpotent Lie algebra, we consider the space of all the derivations such that the corresponding solvable extension has a metric with negative Ricci curvature. Using the nice convexity properties of the moment map for the variety of nilpotent Lie algebras, we obtain a useful characterization of such derivations and some …
We review some constructions and properties of complex manifolds admitting pluriclosed and balanced metrics. We prove that for a 6-dimensional solvmanifold endowed with an invariant complex structure J having holomorphically trivial canonical bundle the pluriclosed flow has a long time solution for every invariant initial datum. Moreover, we state a new conjecture about the existence of balanced and SKT metrics on compact complex manifolds. We show that the conjecture is true for nilmanifolds of dimension 6 and 8 and for 6-dimensional solvmanifolds with holomorphically trivial canonical bundle…
In this paper we investigate the existence of invariant SKT, balanced and generalized Kähler structures on compact quotients $Γ\backslash G$, where $G$ is an almost nilpotent Lie group whose nilradical has one-dimensional commutator and $Γ$ is a lattice of $G$. We first obtain a characterization of Hermitian almost nilpotent Lie algebras $\mathfrak{g}$ whose nilradical $\mathfrak{n}$ has one-dimensional commutator and a classification result in real dimension six. Then, we study the ones admitting SKT and balanced structures and we examine the behaviour of such structures under flows. In parti…
We compute almost-complex invariants $h^{p,0}_{\overline\partial}$, $h^{p,0}_{\text{Dol}}$ and almost-Hermitian invariants $h^{p,0}_{\barδ}$ on families of almost-Kähler and almost-Hermitian $6$-dimensional solvmanifolds. Finally, as a consequence of almost-Kähler identities we provide an obstruction to the existence of a symplectic structure on a given compact almost-complex manifold. Notice that, when $(X,J,g,ω)$ is a compact almost Hermitian manifold of real dimension greater than four, not much is known concerning the numbers $h^{p,q}_{\overline\partial}$.
The general aim of this paper is to study which are the solvable Lie groups admitting an Einstein left invariant metric. The space N of all nilpotent Lie brackets on R^n parametrizes a set of (n+1)-dimensional rank-one solvmanifolds, containing the set of all those which are Einstein in that dimension. The moment map for the natural GL(n)-action on N evaluated in a point of N encodes geometric information on the corresponding solvmanifold, allowing us to use strong and well-known results from geometric invariant theory. For instance, the functional on N whose critical points are precisely the …
The structure of a solvable Lie groups admitting an Einstein left-invariant metric is, in a sense, completely determined by the nilradical of its Lie algebra. We give an easy-to-check necessary and sufficient condition for a nilpotent algebra to be an Einstein nilradical whose Einstein derivation has simple eigenvalues. As an application, we classify filiform Einstein nilradicals (modulo known classification results on filiform graded Lie algebras).
A Hermitian structure on a manifold is called locally conformally Kähler (LCK) if it locally admits a conformal change which is Kähler. In this survey we review recent results of invariant LCK structures on solvmanifolds and present original results regarding the canonical bundle of solvmanifolds equipped with a Vaisman structure, that is, a LCK structure whose associated Lee form is parallel.
We study the existence of strong Kähler with torsion (SKT) metrics and of symplectic forms taming invariant complex structures $J$ on solvmanifolds $G/Γ$ providing some negative results for some classes of solvmanifolds. In particular, we show that if either $J$ is invariant under the action of a nilpotent complement of the nilradical of $G$ or $J$ is abelian or $G$ is almost abelian (not of type (I)), then the solvmanifold $G/Γ$ cannot admit any symplectic form taming the complex structure $J$, unless $G/Γ$ is Kähler. As a consequence, we show that the family of non-Kähler complex manifolds c…
The only known examples of noncompact Einstein homogeneous spaces are standard solvmanifolds (special solvable Lie groups endowed with a left invariant metric), and according to a long standing conjecture, they might be all. The classification of Einstein solvmanifolds is equivalent to the one of Einstein nilradicals, i.e. nilpotent Lie algebras which are nilradicals of the Lie algebras of Einstein solvmanifolds. Up to now, there have been found very few examples of graded nilpotent Lie algebras that can not be Einstein nilradicals. In particular, in each dimension, there are only finitely man…
Let $G=\C^{n}\ltimes_ϕ \C^{m}$ with a semi-simple action $ϕ: \C^{n}\to GL_{m}(\C)$ (not necessarily holomorphic). Suppose $G$ has a lattice $Γ$. Then we show that in some conditions on $G$ and $Γ$, $G/Γ$ admits a Hermitian metric such that the space of harmonic forms satisfies the Hodge symmetry and decomposition. By this result we give many examples of non-Kähler Hermitian solvmanifolds satisfying the Hodge symmetry and decomposition.
A $n$-dimensional Lie group $G$ equipped with a left invariant symplectic form $\om^+$ is called a symplectic Lie group. It is well-known that $\om^+$ induces a left invariant affine structure on $G$. Relatively to this affine structure we show that the left invariant Poisson tensor $π^+$ corresponding to $\om^+$ is polynomial of degree 1 and any right invariant $k$-multivector field on $G$ is polynomial of degree at most $k$. If $G$ is unimodular, the symplectic form $\om^+$ is also polynomial and the volume form $\wedge^{\frac{n}2}\om^+$ is parallel. We show also that any left invariant tens…
For a simply connected (non-nilpotent) solvable Lie group $G$ with a lattice $Γ$ the de Rham and Dolbeault cohomologies of the solvmanifold $G/Γ$ are not in general isomorphic to the cohomologies of the Lie algebra $\mathfrak g$ of $G$. In this paper we construct, up to a finite group, a new Lie algebra $\tilde{\mathfrak g}$ whose cohomology is isomorphic to the de Rham cohomology of $G/Γ$ by using a modification of $G$ associated with a algebraic sub-torus of the Zariski-closure of the image of the adjoint representation. This technique includes the construction due to Guan and developed by t…
Starting from a 6-dimensional nilpotent Lie group N endowed with an invariant SU(3) structure, we construct a homogeneous conformally parallel G_2-metric on an associated solvmanifold. We classify all half-flat SU(3) structures that endow the rank-one solvable extension of N with a conformally parallel G_2 structure. By suitably deforming the SU(3) structures obtained, we are able to describe the corresponding non-homogeneous Ricci-flat metrics with holonomy contained in G_2. In the process we also find a new metric with exceptional holonomy.
We show that the Frölicher spectral sequence of a complex parallelizable solvmanifold is degenerate at $E_{2}$-term. For a semi-direct product $G=\C^{n}\ltimes_ϕN$ of Lie-groups with lattice $Γ=Γ^{\prime}\ltimes Γ^{\prime\prime}$ 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 Frölicher spectral sequence of the nilmanifold $N/Γ^{\prime\prime}$ is degenerate at $E_{r}$-term for $r\ge 2$, then the Frölicher spectral sequence of the solvmanifold $G/Γ$ is also degenerate at $E_{r}$-term.
We study conditions under which sub-complexes of a double complex of vector spaces allow to compute the Bott-Chern cohomology. We are especially aimed at studying the Bott-Chern cohomology of special classes of solvmanifolds, namely, complex parallelizable solvmanifolds and solvmanifolds of splitting type. More precisely, we can construct explicit finite-dimensional double complexes that allow to compute the Bott-Chern cohomology of compact quotients of complex Lie groups, respectively, of some Lie groups of the type $\mathbb{C}^n\ltimes_φN$ where $N$ is nilpotent. As an application, we comput…
We study the positive Hermitian curvature flow of left-invariant metrics on complex 2-step nilpotent Lie groups. In this setting we completely characterize the long-time behaviour of the flow, showing that normalized solutions to the flow subconverge to a non-flat algebraic soliton, in Cheeger- Gromov topology. We also exhibit a uniqueness result for algebraic solitons on such Lie groups.
$p$-Kähler structures on nilmanifolds and holomorphically parallelizable manifolds
Dipartimento di Scienze Matematiche, Fisiche e Informatiche Unità di Matematica e Informatica Università degli Studi di Parma（帕尔马大学数学、物理与信息科学系数学与信息单元）；Dipartimento di Matematica e Informatica Universita degli Studi di Ferrara（费拉拉大学数学与信息学系）；IMAR（罗马尼亚数学研究所）
We study the positive Hermitian curvature flow on the space of left-invariant metrics on complex Lie groups. We show that in the nilpotent case, the flow exists for all positive times and subconverges in the Cheeger-Gromov sense to a soliton. We also show convergence to a soliton when the complex Lie group is almost abelian. That is, when its Lie algebra admits a (complex) co-dimension one abelian ideal. Finally, we study solitons in the almost-abelian setting. We prove uniqueness and completely classify all left-invariant, almost-abelian solitons, giving a method to construct examples in arbi…
Let $G$ be a simply connected solvable Lie group with a lattice $Γ$ and the Lie algebra $\g$ and a representation $ρ:G\to GL(V_ρ)$ whose restriction on the nilradical is unipotent. Consider the flat bundle $E_ρ$ given by $ρ$. By using "many" characters $\{α\}$ of $G$ and "many" flat line bundles $\{E_α\}$ over $G/Γ$, we show that an isomorphism \[\bigoplus_{\{α\}} H^{\ast}(\g, V_α\otimes V_ρ)\cong \bigoplus_{\{E_α\}} H^{\ast}(G/Γ, E_α\otimes E_ρ)\] holds. This isomorphism is a generalization of the well-known fact:"If $G$ is nilpotent and $ρ$ is unipotent then, the isomorphism $H^{\ast}(\g, V_…
We study nilpotent Lie algebras endowed with a complex structure and a quadratic structure which is pseudo-Hermitian for the given complex structure. We propose several methods to construct such Lie algebras and describe a method of double extension by planes to get an inductive description of all of them. As an application, we give a complete classification of nilpotent quadratic Lie algebras where the metric is Lorentz-Hermitian and we fully classify all nilpotent pseudo-Hermitian quadratic Lie algebras up to dimension 8 and their inequivalent pseudo-Hermitian metrics.
We study the six-dimensional solvmanifolds that admit complex structures of splitting type classifying the underlying solvable Lie algebras. In particular, many complex structures of this type exist on the Nakamura manifold $X$, and they allow us to construct a countable family of compact complex non-$\partial\overline\partial$ manifolds $X_k$, $k\in\mathbb{Z}$, that admit a small holomorphic deformation $\{(X_{k})_{t}\}_{t\inΔ_k}$ satisfying the $\partial\overline\partial$-Lemma for any $t\inΔ_k$ except for the central fibre. Moreover, a study of the existence of special Hermitian metrics is …
The study of quasi-Kähler Chern-flat almost Hermitian manifolds is strictly related to the study of anti-bi-invariant almost complex Lie algebras. In the present paper we show that quasi-Kähler Chern-flat almost Hermitian structures on compact manifolds are in correspondence to complex parallelisable Hermitian structures satisfying the second Gray identity. From an algebraic point of view this correspondence reads as a natural correspondence between anti-bi-invariant almost complex structures on Lie algebras to bi-invariant complex structures. Some natural algebraic problems are approached and…
The problem of classifying Einstein solvmanifolds, or equivalently, Ricci soliton nilmanifolds, is known to be equivalent to a question on the variety of n-dimensional complex nilpotent Lie algebra laws. Namely, one has to determine which GL(n)-orbits in this variety have a critical point of the squared norm of the moment map. In dimension 7, there are 148 complex nilpotent Lie algebras and 6 curves of pairwise non-isomorphic nilpotent Lie algebras, and we give in this paper a complete classification of the aforementioned distinguished orbits.
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 by A. Fino and L. Vezzoni that a compact complex manifold admitting both a balanced metric and a 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 by L. Bedull…
This paper is concerned with Chern-Ricci flow evolution of left-invariant hermitian structures on Lie groups. We study the behavior of a solution, as t is approaching the first time singularity, by rescaling in order to prevent collapsing and obtain convergence in the pointed (or Cheeger-Gromov) sense to a Chern-Ricci soliton. We give some results on the Chern-Ricci form and the Lie group structure of the pointed limit in terms of the starting hermitian metric and, as an application, we obtain a complete picture for the class of solvable Lie groups having a codimension one normal abelian subgr…
We exhibit Osserman metrics with non-nilpotent Jacobi operators and with non-trivial Jordan normal form in neutral signature (n,n) for any n which is at least 3. These examples admit a natural almost para-Hermitian structure and are semi para-complex Osserman with non-trivial Jordan normal form as well; they neither satisfy the third Gray identity nor are they integrable.
In this paper we determine the moduli space, up to isometric automorphism, of left-invariant metrics on a $6$-dimensional Lie group $H$, such that its Lie algebra $\mathfrak{h}$ admits a complex structure and has first Betti number equal to four. We also investigate which of these metrics are Hermitian and classify the corresponding complex structures.
A Hermitian metric on a complex manifold is called SKT (strong Kähler with torsion) if the Bismut torsion $3$-form $H$ is closed. As the conformal generalization of the SKT condition, we introduce a new type of Hermitian structure, called \emph{locally conformal SKT} (or shortly LCSKT). More precisely, a Hermitian structure $(J,g)$ is said to be LCSKT if there exists a closed non-zero $1$-form $α$ such that $d H = α\wedge H$. In the paper we consider non-trivial LCSKT structures, i.e. we assume that $d H \neq 0$ and we study their existence on Lie groups and their compact quotients by lattices…
We study curvature flows in the locally homogeneous case (e.g. compact quotients of Lie groups, solvmanifolds, nilmanifolds) in a unified way, by considering a generic flow under just a few natural conditions on the broad class of almost-hermitian structures. As a main tool, we use an ODE system defined on the variety of 2n-dimensional Lie algebras, called the bracket flow, whose solutions differ from those to the original curvature flow by only pull-back by time-dependent diffeomorphisms. The approach, which has already been used to study the Ricci flow on homogeneous manifolds, is useful to …
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-Kahler, 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. New examples of multiparametric families of metrics of signature (3,3) and almost para-complex pseudo-Riemannian half-flat structures on six-dimensional nilmanifold…
Let $\mathfrak{g}$ be a $2n$-dimensional unimodular Lie algebra equipped with a Hermitian structure $(J,F)$ such that the complex structure $J$ is abelian and the fundamental form $F$ is balanced. We prove that the holonomy group of the associated Bismut connection reduces to a subgroup of $SU(n-k)$, being $2k$ the dimension of the center of $\mathfrak{g}$. We determine conditions that allow a unimodular Lie algebra to admit this particular type of structures. Moreover, we give methods to construct them in arbitrary dimensions and classify them if the Lie algebra is 8-dimensional and nilpotent…
We obtain some general results on Sasakian Lie algebras and prove as a consequence that a (2n + 1)-dimensional nilpotent Lie group admitting left-invariant Sasakian structures is isomorphic to the real Heisenberg group $H_{2n + 1}$. Furthermore, we classify Sasakian Lie algebras of dimension 5 and determine which of them carry a Sasakian $α$-Einstein structure. We show that a 5-dimensional solvable Lie group with a left-invariant Sasakian structure and which admits a compact quotient by a discrete subgroup is isomorphic to either $H_5$ or a semidirect product $\R \ltimes (H_3 \times \R)$. In p…
We study left-invariant generalized Kähler 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 Kähler almost abelian Lie groups and determine the 6-dimensional compact almost abelian solvmanifolds admitting an invariant generali…
We show that the existence of a left-invariant pluriclosed Hermitian metric on a unimodular Lie group with a left-invariant abelian complex structure forces the group to be $2$-step nilpotent. Moreover, we prove that the pluriclosed flow starting from a left-invariant Hermitian metric on a $2$-step nilpotent Lie group preserves the Strominger Kähler-like condition.
We study hermitian structures, with respect to the standard neutral metric on the cotangent bundle $T^*G$ of a 2n-dimensional Lie group $G$, which are left invariant with respect to the Lie group structure on $T^*G$ induced by the coadjoint action. These are in one-to-one correspondence with left invariant generalized complex structures on $G$. Using this correspondence and results of Cavalcanti-Gualtieri and Fernández-Gotay-Gray, it turns out that when $G$ is nilpotent and four or six dimensional, the cotangent bundle $T^*G$ always has a hermitian structure. However, we prove that if $G$ is a…
We investigate the Hermitian curvature flow (HCF) of left-invariant metrics on complex unimodular Lie groups. We show that in this setting the flow is governed by the Ricci-flow type equation $\partial_tg_{t}=-{\rm Ric}^{1,1} (g_t)$. The solution $g_t$ always exist for all positive times, and $(1 + t)^{-1}g_t$ converges as $t\to \infty$ in Cheeger-Gromov sense to a non-flat left-invariant soliton $(\bar G, \bar g)$. Moreover, up to homotheties on each of these groups there exists at most one left-invariant soliton solution, which is a static Hermitian metric if and only if the group is semisim…
In this paper we study a version of the Hermitian curvature flow (HCF). We focus on complex homogeneous manifolds equipped with induced metrics. We prove that this finite-dimensional space of metrics is invariant under the HCF and write down the corresponding ODE on the space of Hermitian forms on the underlying Lie algebra. Using these computations we construct HCF-Einstein metrics on $G$-homogeneous manifolds, where $G$ is a complexification of a compact simple Lie group. We conjecture that under the HCF any induced metric on such a manifold pinches towards the HCF-Einstein metric. For a nil…
FaMAF, Universidad Nacional de Córdoba（科尔多瓦国立大学数学与应用物理研究所）；Ciem-Conicet（阿根廷国家科学研究委员会数学研究所）；Departamento de Matemática Aplicada, Instituto de Matemática e Estatística, Universidade Federal Fluminense（弗鲁米嫩塞联邦大学数学与统计学院应用数学系）；PGMat, Instituto de Matemática e Estatística, Universidade Federal Fluminense（弗鲁米嫩塞联邦大学数学与统计学院研究生数学项目）
We find a one-parameter family of non-isomorphic nilpotent Lie algebras $\mathfrak{g}_a$, with $a \in [0,\infty)$, 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 $8$-dimensional nilmanifolds admitting a complex structure. Moreover, balanced Hermitian metrics and generalized Gauduchon metrics on such nilmanifolds are constructed.
