Among closed G2-structures there are two very distinguished classes: Laplacian solitons and Extremally Ricci-pinched G2-structures. We study the existence problem and explore possible interplays between these concepts in the context of left-invariant G2-structures on solvable Lie groups. Also, some Ricci pinching properties of G2-structures on solvmanifolds are obtained, in terms of the extremal values and points of the Ricci pinching functional F=scal/|Ric|. Many natural open problems have been included.
In this paper we study some properties of almost abelian solvmanifolds using minimal models associated to a fibration. In particular we state a necessary and sufficient condition to formality and a method for finding symplectic strucures of this kind of solvmanifolds.
We study the $G_2$ analogue of the Goldberg conjecture on non-compact solvmanifolds. In contrast to the almost-Kähler case we prove that a 7-dimensional solvmanifold cannot admit any left-invariant calibrated $G_2$-structure $φ$ such that the induced metric $g_φ$ is Einstein, unless $g_φ$ is flat. We give an example of 7-dimensional solvmanifold admitting a left-invariant calibrated $G_2$-structure $φ$ such that $g_φ$ is Ricci-soliton. Moreover, we show that a 7-dimensional (non-flat) Einstein solvmanifold $(S,g)$ cannot admit any left-invariant cocalibrated $G_2$-structure $φ$ such that the i…
We construct a compact 6-dimensional solvmanifold endowed with a non-trivial invariant generalized Kähler structure and which does not admit any Kähler metric. This is in contrast with the case of nilmanifolds which cannot admit any invariant generalized Kähler structure unless they are tori.
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.
A compact solvmanifold of completely solvable type, i.e. a compact quotient of a completely solvable Lie group by a lattice, has a Kähler structure if and only if it is a complex torus. We show more in general that a compact solvmanifold $M$ of completely solvable type endowed with an invariant complex structure $J$ admits a symplectic form taming J if and only if $M$ is a complex torus. This result generalizes the one obtained in [7] for nilmanifolds.
In this article we study the relation between flat solvmanifolds and $G_2$-geometry. First, we give a classification of 7-dimensional flat splittable solvmanifolds using the classification of finite subgroups of $\mathsf{GL}(n,\mathbb{Z})$ for $n=5$ and $n=6$. Then, we look for closed, coclosed and divergence-free $G_2$-structures compatible with the flat metric on them. In particular, we provide explicit examples of compact flat manifolds with a torsion-free $G_2$-structure whose finite holonomy is cyclic and contained in $G_2$, and examples of compact flat manifolds admitting a divergence-fr…
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…
We obtain new examples of non-symmetric Einstein solvmanifolds by combining two techniques. In \cite{T2}, H. Tamaru constructs new {\em attached} solvmanifolds, which are submanifolds of the solvmanifolds corresponding to noncompact symmetric spaces, endowed with a natural metric. Extending this construction, we apply it to {\em associated} solvmanifolds, described in \cite{GK}, obtained by modifying the algebraic structure of the solvable Lie algebras corresponding to noncompact symmetric spaces. Our new examples are Einstein solvmanifolds with nilradicals of high nilpotency, which are geomet…
We characterize unimodular solvable Lie algebras with Vaisman structures in terms of Kähler flat Lie algebras equipped with a suitable derivation. Using this characterization we obtain algebraic restrictions for the existence of Vaisman structures and we establish some relations with other geometric notions, such as Sasakian, coKähler and left-symmetric algebra structures. Applying these results we construct families of Lie algebras and Lie groups admitting a Vaisman structure and we show the existence of lattices in some of these families, obtaining in this way many examples of new solvmanifo…
We classify solvable Lie groups admitting left invariant symplectic half-flat structure. When the Lie group has a compact quotient by a lattice, we show that these structures provide solutions of supersymmetric equations of type IIA.
We determine the 6-dimensional solvmanifolds admitting an invariant complex structure with holomorphically trivial canonical bundle. Such complex structures are classified up to isomorphism, and the existence of strong Kähler with torsion (SKT), generalized Gauduchon, balanced and strongly Gauduchon metrics is studied. As an application we construct a holomorphic family $(M,J_a)$ of compact complex manifolds such that $(M,J_a)$ satisfies the $\partial\bar\partial$-Lemma and admits a balanced metric for any $a\not=0$, but the central limit neither satisfies the $\partial\bar\partial$-Lemma nor …
We discuss our recent results on the existence and classification problem of complex and Kaehler structures on compact solvmanifolds. In particular, we determine in this paper all the complex surfaces which are diffeomorphic to compact solvmanifolds (and compact homogeneous manifolds in general).
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).
Special structures on almost abelian solvmanifolds
Institute of Mathematics “Simion Stoilow” of the Romanian Academy（罗马尼亚科学院西米恩·斯托伊洛数学研究所）；Université Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay（巴黎萨克雷大学，法国国家科学研究中心，奥赛数学实验室）
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 cohcain complexes which computes the holomorphic Poisson cohomology of nilmanifolds and solvmanifolds.
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.
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…
A solvmanifold is a compact differentiable manifold M on which a connected solvable Lie group G acts transitively. As the main result, we will see, applying a result of Arapura and Nori on solvable Kaehler groups and some of the author's previous results, that a compact solvmanifold admits a Kaehler structure if and only if it is a finite quotient of a complex torus which has a structure of a complex torus bundle over a complex torus. There are also quite a few comments and remarks on and around this result.
We classify four-dimensional compact solvmanifolds up to diffeomorphism, while determining which of them have complex analytic structures. In particular, we shall see that a four-dimensional compact solvmanifold S can be written, up to double covering, as G/L where G is a simply connected solvable Lie group and L is a lattice of G, and every complex structure J on S is the canonical complex structure induced from a left-invariant complex structure on G. We also give a complete list of all the complex structures on four-dimensional compact homogeneous spaces, referring to their corresponding co…
We study Lie algebras of type I, that is, a Lie algebra $\mathfrak{g}$ where all the eigenvalues of the operator ad$_X$ are imaginary for all $X\in \mathfrak{g}$. We prove that the Morse-Novikov cohomology of a Lie algebra of type I is trivial for any closed $1$-form. We focus on locally conformal symplectic structures (LCS) on Lie algebras of type I. In particular we show that for a Lie algebra of type I any LCS structure is of the first kind. We also exhibit lattices for some $6$-dimensional Lie groups of type I admitting left invariant LCS structures in order to produce compact solvmanifold…
The aim of this paper is to continue the study of Kodaira dimension for almost complex manifolds, focusing on the case of compact $4$-dimensional solvmanifolds without any integrable almost complex structure. According to the classification theory we consider: $\mathfrak{r}\mathfrak{r}_{3, -1}$, $\mathfrak{nil}^4$ and $\mathfrak{r}_{4, λ, -(1 + λ)}$ with $-1 < λ< -\frac{1}{2}$. For the first solvmanifold we introduce special families of almost complex structures, compute the corresponding Kodaira dimension and show that it is no longer a deformation invariant. Moreover we prove Ricci flatness …
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…
We show that a 7-dimensional non-compact Ricci-flat Riemannian manifold with Riemannian holonomy G_2 can admit non-integrable G_2 structures of type R + S^2_0(R^7) + R^7 in the sense of Fernández and Gray. This relies on the construction of some G_2 solvmanifolds, whose Levi-Civita connection is known to give a parallel spinor, admitting a 2-parameter family of metric connections with non-zero skew-symmetric torsion that has parallel spinors as well. The family turns out to be a deformation of the Levi-Civita connection. This is in contrast with the case of compact scalar-flat Riemannian spin …
$p$-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 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 …
We construct new homogeneous Einstein spaces with negative Ricci curvature in two ways: First, we give a method for classifying and constructing a class of rank one Einstein solvmanifolds whose derived algebras are two-step nilpotent. As an application, we describe an explicit continuous family of ten-dimensional Einstein manifolds with a two-dimensional parameter space, including a continuous subfamily of manifolds with negative sectional curvature. Secondly, we obtain new examples of non-symmetric Einstein solvmanifolds by modifying the algebraic structure of non-compact irreducible symmetri…
On the generalized cohomologies of almost abelian solvmanifolds
Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas（坎皮纳斯州立大学数学、统计与科学计算研究所）；Institute of Mathematics and Informatics, Bulgarian Academy of Sciences（保加利亚科学院数学与信息学研究所）
In this paper we consider closed $\text{SL}(3,\mathbb{C})$-structures which are either mean convex or tamed by a symplectic form. These notions were introduced by Donaldson in relation to $\text{G}_2$-manifolds with boundary. In particular, we classify nilmanifolds which carry an invariant mean convex closed $\text{SL}(3,\mathbb{C})$-structure and those which admit an invariant mean convex half-flat $\text{SU}(3)$-structure. We also prove that, if a solvmanifold admits an invariant tamed closed $\text{SL}(3,\mathbb{C})$-structure, then it also has an invariant symplectic half-flat $\text{SU}(3…
We prove a structure theorem for homogeneous closed gradient Laplacian solitons and use it to show some examples of closed Laplacian solitons cannot be made gradient. More specifically, we show that the Laplacian solitons on nilpotent Lie groups found by Nicolini are not gradient up to homothetic $G_2$-structures except for $N_1$, where $f$ must be a Gaussian. We also show that the closed $G_2$-structure $φ_{12}$ on $N_{12}$ constructed by Fernández-Fino-Manero cannot be a gradient soliton. We further show that closed non-torsion-free gradient Laplacian solitons on almost abelian solvmanifolds…
A study is made of left-invariant $\mathrm{G}_2$-structures with an exact 3-form on a Lie group $G$ whose Lie algebra $\mathfrak{g}$ admits a codimension-one nilpotent ideal $\mathfrak{h}$. It is shown that such a Lie group $G$ cannot admit a left-invariant closed $\mathrm{G}_2$-eigenform for the Laplacian and that any compact solvmanifold $Γ\backslash G$ arising from $G$ does not admit an (invariant) exact $\mathrm{G}_2$-structure. We also classify the seven-dimensional Lie algebras $\mathfrak{g}$ with codimension-one ideal equal to the complex Heisenberg Lie algebra which admit exact $\mathr…
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…
This note summarizes the talk by the author at the workshop "Geometry and Computer Science" held in Pescara in February 2017. We present how SageMath can help in research in Complex and Differential Geometry, with two simple applications, which are not intended to be original. We consider two "classification problems" on quotients of Lie groups, namely, "computing cohomological invariants" [D. Angella, M. G. Franzini, F. A. Rossi, Degree of non-Kählerianity for 6-dimensional nilmanifolds, Manuscripta Math. 148 (2015), no. 1-2, 177--211], [A. Latorre, L. Ugarte, R. Villacampa, On the Bott-Chern…
We bring new insights into the long-standing Alekseevskii conjecture, namely that any connected homogeneous Einstein manifold of negative scalar curvature is diffeomorphic to a Euclidean space, by proving structural results which are actually valid for any homogeneous expanding Ricci soliton, and generalize many well-known results on Einstein solvmanifolds, solvsolitons and nilsolitons. We obtain that any homogeneous expanding Ricci soliton M=G/K is diffeomorphic to a product U/K x N, where U is a maximal reductive Lie subgroup of G and N is the maximal nilpotent normal subgroup of G, such tha…
We propose studies of special Riemannian geometries with structure groups $H_1=SO(3)\subset SO(5)$, $H_2=SU(3)\subset SO(8)$, $H_3=Sp(3)\subset SO(14)$ and $H_4=F_4\subset SO(26)$ in respective dimensions 5, 8, 14 and 26. These geometries, have torsionless models with symmetry groups $G_1=SU(3)$, $G_2=SU(3)\times SU(3)$, $G_3=SU(6)$ and $G_4=E_6$. The groups $H_k$ and $G_k$ constitute a part of the `magic square' for Lie groups. Apart from the $H_k$ geometries in dimensions $n_k$, the `magic square' Lie groups suggest studies of a finite number of other special Riemannian geometries. Among the…
We introduce a class of first order G-structures, each of which has an underlying almost conformally symplectic structure. There is one such structure for each real simple Lie algebra which is not of type $C_n$ and admits a contact grading. We show that a structure of each of these types on a smooth manifold $M$ determines a canonical compatible linear connection on the tangent bundle $TM$. This connection is characterized by a normalization condition on its torsion. The algebraic background for this result is proved using Kostant's theorem on Lie algebra cohomology. For each type, we give an …
It is shown that the compact Lie group SU(3) admits an Sp(2)Sp(1)-structure whose distinguished 2-forms $ω_1,ω_2,ω_3$ span a differential ideal. This is achieved by first reducing the structure further to a subgroup isomorphic to SO(3).
In all odd dimensions $\geq 5$ we produce examples of manifolds admitting pairs of Sasaki structures with different basic Hodge numbers. In dimension $5$ we prove more precise results, for example we show that on connected sums of copies of $S^2\times S^3$ the number of Sasaki structures with different basic Hodge numbers within a fixed homotopy class of almost contact structures is unbounded. All the Sasaki structures we consider are negative in the sense that the basic first Chern class is represented by a negative definite form of type $(1,1)$. We also discuss the relation of these results …
Let N be a nilpotent Lie group and let S be an invariant geometric structure on N (cf. symplectic, complex or hypercomplex). We define a left invariant Riemannian metric on N compatible with S to be "minimal", if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar curvature. We prove that minimal metrics (if any) are unique up to isometry and scaling, they develop soliton solutions for the invariant Ricci flow and are characterized as the critical points of a natural variational problem. The uniqueness allows us to distinguish geome…
We define a CR structure on a distinguished hyperplane in $\mathbb{C}^{n+1}$ and the CR sub-Laplacian on this CR manifold. We also define symmetries of the CR sub-Laplacian in general and for this special case construct all of them using the ambient construction. Then we investigate the algebra structure of the symmetry algebra of the sub-Laplacian. For this purpose we derive the decomposition of $S^{k}_{0}\mathfrak{sl}(V)$ under the action of $SL(V)$.
