arXiv subjects

Adrián Andrada

Publications and source records attributed to Adrián Andrada.

At least 19 recordsLinked to original sources

arXiv2026

Betti and Hodge numbers of solvmanifolds arising from integer polynomials

We study the de Rham cohomology of three families of completely solvable almost abelian solvmanifolds (called basic, complex, and hypercomplex) constructed from a monic integer polynomial with positive distinct roots whose product equals 1, following the work of Andrada and Barberis. Under two algebraic restrictions on such polynomials (the full rank and quasi full rank conditions) we compute the Betti numbers and Poincaré polynomials of these manifolds. Moreover, we study the Dolbeault cohomology of the complex solvmanifolds by identifying them with generalized Nakamura manifolds recently introduced by Cattaneo and Tomassini. Assuming a suitable condition on the lattice, we compute their Hodge numbers, which exhibit a combinatorial structure related to Pascal's triangle in the full rank setting, and are described by explicit generating polynomials in the quasi full rank case.

math.DG↗

arXiv2026

Complex structures on 2-step nilpotent Lie algebras arising from graphs

This work investigates the existence of complex structures on 2-step nilpotent Lie algebras arising from finite graphs. We introduce the notion of adapted complex structure, namely a complex structure that maps vertices and edges of the graph to vertices and edges, and we analyze in depth the restrictions imposed by the integrability condition. We completely characterize the graphs that admit abelian adapted complex structures, showing that they belong to a small family of graphs that we call basic. We prove that any graph endowed with an adapted complex structure $J$ contains a unique $J$-invariant basic spanning subgraph, and conversely, that every such graph can be constructed through a systematic expansion procedure starting from a basic graph. We also explore geometric and combinatorial consequences, including the existence of special Hermitian metrics as well as other graph-theoretic properties.

math.DG↗

arXiv2026

Holonomy of the Obata connection on 2-step hypercomplex nilmanifolds

We study the holonomy of the Obata connection on 2-step hypercomplex nilmanifolds. By explicitly computing the curvature tensor, we determine the conditions under which the Obata connection is flat, showing that this depends on the nilpotency step of each complex structure. In particular, we show that for 2-step hypercomplex nilmanifolds the holonomy algebra of the Obata connection is always an abelian subalgebra of $\mathfrak{sl}(n, \mathbb{H})$ and we prove that the $\mathbb{H}$-solvable conjecture holds in this case. Furthermore, we provide new examples of $k$-step nilpotent hypercomplex nilmanifolds, with arbitrary $k$, which are not Obata flat.

math.DG↗

arXiv2026

1-Lefschetz contact solvmanifolds

We study the contact Lefschetz condition on compact contact solvmanifolds, as introduced by B.\ Cappelletti-Montano, A.\ De Nicola and I.\ Yudin. We seek to fill the gap in the literature concerning Benson-Gordon type results, characterizing $1$-Lefschetz contact solvmanifolds. We prove that the $1$-Lefschetz condition on Lie algebras is preserved via $1$-dimensional central extensions by a symplectic cocycle, thereby establishing that a unimodular symplectic Lie algebra $(\mathfrak{h}, ω)$ is $1$-Lefschetz if and only if its contactization $(\mathfrak{g}, η)$ is $1$-Lefschetz. We achieve this by showing an explicit relation for the relevant cohomology degrees of $\mathfrak{h}$ and $\mathfrak{g}$. Using this, we show how the commutators $[\mathfrak{h},\mathfrak{h}]$ and $[\mathfrak{g},\mathfrak{g}]$ are related, especially when the $1$-Lefschetz condition holds. By specializing to the nilpotent setting, we prove that $1$-Lefschetz contact nilmanifolds equipped with an invariant contact form are quotients of a Heisenberg group, and deduce that there are many examples of compact $K$-contact solvmanifolds not admitting compatible Sasakian structures. We also construct examples of completely solvable $1$-Lefschetz solvmanifolds, some having the $2$-Lefschetz property and some failing it.

math.DG↗

arXiv2025

Symplectic solvmanifolds not satisfying the hard-Lefschetz condition

For Lie groups $G$ of the form $G = \R^k \ltimes_ϕ \R^m$, with $k + m$ even, a result of H. Kasuya shows that if the action $ϕ:\R^k \to \mathrm{Aut}(\R^m)$ is semisimple then any symplectic solvmanifold $(Γ\backslash G, ω)$ satisfies the hard-Lefschetz condition for any symplectic form. In this article, we prove the converse in the case $k = 1$ and $G$ completely solvable: no symplectic form on such a solvmanifold satisfies the hard-Lefschetz condition if $ϕ$ is not semisimple; moreover, we show that the failure occurs either at degree $1$ or at degree $2$ in cohomology, depending on the spectrum of the differential of the action $ϕ$. This result is achieved through a detailed analysis of the cohomology groups $H^1(\g)$, $H^2(\g)$, $H^{2n-2}(\g)$, $H^{2n-1}(\g)$ of the Lie algebra $\g$ of such Lie groups. Among other things, this analysis yields useful representatives for each cohomology class corresponding to any symplectic form on $\g$, allowing the most delicate cases to be reduced to a straightforward computation. We also construct lattices for many of the Lie groups under consideration, thereby exhibiting examples of symplectic solvmanifolds of completely solvable Lie groups failing to have the hard-Lefschetz property for any symplectic form.

math.DG↗

arXiv2025

Hypercomplex structures on special linear groups

The purpose of this article is twofold. First, we prove that the $8$-dimensional Lie group $\operatorname{SL}(3,\mathbb{R})$ does not admit a left-invariant hypercomplex structure. To accomplish this we revise the classification of left-invariant complex structures on $\operatorname{SL}(3,\mathbb{R})$ due to Sasaki. Second, we exhibit a left-invariant hypercomplex structure on $\operatorname{SL}(2n+1,\mathbb{C})$, which arises from a complex product structure on $\operatorname{SL}(2n+1,\mathbb{R})$, for all $n\in \mathbb{N}$. We then show that there are no HKT metrics compatible with this hypercomplex structure. Additionally, we determine the associated Obata connection and we compute explicitly its holonomy group, providing thus a new example of an Obata holonomy group properly contained in $\operatorname{GL}(m,\mathbb{H})$ and not contained in $\operatorname{SL}(m,\mathbb{H})$, where $4m=\dim_\mathbb{R} \operatorname{SL}(2n+1,\mathbb{C})$.

math.DG↗

arXiv2025

Almost abelian complex nilmanifolds

We show that a complex structure on a nilpotent almost abelian real Lie algebra is unique if it exists. As a consequence, we get full control over the cohomology and deformations of almost abelian complex nilmanifolds.

math.DG↗

arXiv2024

Construction of symplectic solvmanifolds satisfying the hard-Lefschetz condition

A compact symplectic manifold $(M, ω)$ is said to satisfy the hard-Lefschetz condition if it is possible to develop an analogue of Hodge theory for $(M, ω)$. This loosely means that there is a notion of harmonicity of differential forms in $M$, depending on $ω$ alone, such that every de Rham cohomology class in has a $ω$-harmonic representative. In this article, we study two non-equivalent families of diagonal almost-abelian Lie algebras that admit a distinguished almost-Kähler structure and compute their cohomology explicitly. We show that they satisfy the hard-Lefschetz condition with respect to any left-invariant symplectic structure by exploiting an unforeseen connection with Kneser graphs. We also show that for some choice of parameters their associated simply connected, completely solvable Lie groups admit lattices, thereby constructing examples of almost-Kähler solvmanifolds satisfying the hard-Lefschetz condition, in such a way that their de Rham cohomology is fully known.

math.DG↗

arXiv2024

Applications of the quaternionic Jordan form to hypercomplex geometry

We apply the quaternionic Jordan form to classify the hypercomplex nilpotent almost abelian Lie algebras in all dimensions and to carry out the complete classification of 12-dimensional hypercomplex almost abelian Lie algebras. Moreover, we determine which 12-dimensional simply connected hypercomplex almost abelian Lie groups admit lattices. Finally, for each integer $n>1$ we construct infinitely many, up to diffeomorphism, $(4n+4)$-dimensional hypercomplex almost abelian solvmanifolds which are completely solvable. These solvmanifolds arise from a distinguished family of monic integer polynomials of degree $n$.

math.DG↗

arXiv2024

Harmonic almost complex structures on almost abelian Lie groups and solvmanifolds

An almost abelian Lie group is a solvable Lie group with a codimension-one normal abelian subgroup. We characterize almost Hermitian structures on almost abelian Lie groups where the almost complex structure is harmonic with respect to the Hermitian metric. Also, we adapt the Gray-Hervella classification of almost Hermitian structures to the family of almost abelian Lie groups. We provide several examples of harmonic almost complex structures in different Gray-Hervella classes on some associated compact almost abelian solvmanifolds.

math.DG↗

arXiv2024

On the canonical bundle of complex solvmanifolds and applications to hypercomplex geometry

We study complex solvmanifolds $Γ\backslash G$ with holomorphically trivial canonical bundle. We show that the trivializing section of this bundle can be either invariant or non-invariant by the action of $G$. First we characterize the existence of invariant trivializing sections in terms of the Koszul 1-form $ψ$ canonically associated to $(\mathfrak{g},J)$, where $\mathfrak{g}$ is the Lie algebra of $G$, and we use this characterization to produce new examples of complex solvmanifolds with trivial canonical bundle. Moreover, we provide an algebraic obstruction, also in terms of $ψ$, for a complex solvmanifold to have trivial (or more generally holomorphically torsion) canonical bundle. Finally, we exhibit a compact hypercomplex solvmanifold $(M^{4n},\{J_1,J_2,J_3\})$ such that the canonical bundle of $(M,J_α)$ is trivial only for $α=1$, so that $M$ is not an $\operatorname{SL}(n,\mathbb{H})$-manifold.

math.DG↗

arXiv2024

Locally conformally product structures on solvmanifolds

We study left invariant locally conformally product structures on simply connected Lie groups and give their complete description in the solvable unimodular case. Based on previous classification results, we then obtain the complete list of solvable unimodular Lie algebras up to dimension 5 which carry LCP structures, and study the existence of lattices in the corresponding simply connected Lie groups.

math.DG↗

arXiv2024

Harmonic complex structures and special Hermitian metrics on products of Sasakian manifolds

It is well known that the product of two Sasakian manifolds carries a 2-parameter family of Hermitian structures $(J_{a,b},g_{a,b})$. We show in this article that the complex structure $J_{a,b}$ is harmonic with respect to $g_{a,b}$, i.e. it is a critical point of the Dirichlet energy functional. Furthermore, we also determine when these Hermitian structures are locally conformally Kähler, balanced, strong Kähler with torsion, Gauduchon or $k$-Gauduchon ($k\geq 2$). Finally, we study the Bismut connection associated to $(J_{a,b}, g_{a,b})$ and we provide formulas for the Bismut-Ricci tensor $\operatorname{Ric}^B$ and the Bismut-Ricci form $ρ^B$. We show that these tensors vanish if and only if each Sasakian factor is $η$-Einstein with appropriate constants and we also exhibit some examples fulfilling these conditions, thus providing new examples of Calabi-Yau with torsion manifolds.

math.DG↗

arXiv2023

Hypercomplex almost abelian solvmanifolds

We give a characterization of almost abelian Lie groups carrying left invariant hypercomplex structures and we show that the corresponding Obata connection is always flat. We determine when such Lie groups admit HKT metrics and study the corresponding Bismut connection. We obtain the classification of hypercomplex almost abelian Lie groups in dimension 8 and determine which ones admit lattices. We show that the corresponding 8-dimensional solvmanifolds are nilmanifolds or admit a flat hyper-Kähler metric. Furthermore, we prove that any 8-dimensional compact flat hyper-Kähler manifold is a solvmanifold equipped with an invariant hyper-Kähler structure. We also construct almost abelian hypercomplex nilmanifolds and solvmanifolds in higher dimensions.

math.DG↗

arXiv2020

Vaisman solvmanifolds and relations with other geometric structures

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 solvmanifolds equipped with invariant Vaisman structures.

math.DG↗

arXiv2019

Killing-Yano 2-forms on 2-step nilpotent Lie groups

In this article we show that the only 2-step nilpotent Lie groups which carry a non-degenerate left invariant Killing-Yano 2-form are the complex Lie groups. In the case of 2-step nilpotent complex Lie groups arising from connected graphs, we prove that the space of left invariant Killing-Yano 2-forms is one-dimensional.

math.DG↗

arXiv2018

Locally conformally Kähler solvmanifolds: a survey

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.

math.DG↗

arXiv2016

Conformal Killing 2-forms on 4-dimensional manifolds

We study 4-dimensional simply connected Lie groups $G$ with left-invariant Riemannian metric $g$ admitting non-trivial conformal Killing 2-forms. We show that either the real line defined by such a form is invariant under the group action, or the metric is half conformally flat. In the first case, the problem reduces to the study of invariant conformally Kähler structures, whereas in the second case, the Lie algebra of $G$ belongs (up to homothety) to a finite list of families of metric Lie algebras.

math.DG↗