arXiv subjects

Agustín Garrone

Publications and source records attributed to Agustín Garrone.

6 recordsLinked to original sources

arXiv2026

On centerless unimodular contact Lie algebras

We provide an elementary proof that, in a (transversely) unimodular contact Lie algebra, the adjoint action of the Reeb vector is nilpotent except when the Lie algebra is isomorphic to either $\mathfrak{sl}(2,\mathbb{R})$ or $\mathfrak{su}(2)$. We introduce a class of contact Lie algebras, called \textit{DS-contact Lie algebras}, containing all K-contact Lie algebras, and deduce from the previous result that the only centerless unimodular examples in this class are precisely $\mathfrak{sl}(2,\mathbb{R})$ and $\mathfrak{su}(2)$. This gives an alternative proof of the previously known fact that centerless unimodular Sasakian Lie algebras are isomorphic to either $\mathfrak{sl}(2,\mathbb{R})$ or $\mathfrak{su}(2)$. Some other results known to hold for Sasakian Lie algebras are generalized as well. We investigate several properties of DS-contact Lie algebras in relation to Frobenius Lie algebras, and also classify them in dimension five. Some implications for the contact Lefschetz condition are explored.

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↗

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

New examples of G$_2$-structures with divergence-free torsion

Interest in Riemannian manifolds with holonomy equal to the exceptional Lie group $\mathrm{G}_2$ have spurred extensive research in geometric flows of $\mathrm{G}_2$-structures defined on seven-dimensional manifolds in recent years. Among many possible geometric flows, the so-called \textit{isometric flow} has the distinctive feature of preserving the underlying metric induced by that $\mathrm{G}_2$-structure, so it can be used to evolve a $\mathrm{G}_2$-structure to one with the smallest possible torsion in a given metric class. This flow is built upon the divergence of the full torsion tensor of the flowing $\mathrm{G}_2$-structures in such a way that its critical points are precisely $\mathrm{G}_2$-structures with divergence-free torsion. In this article we study three large families of pairwise non-equivalent non-closed left-invariant $\mathrm{G}_2$-structures defined on simply connected solvable Lie groups previously studied in \cite{KL} and compute the divergence of their full torsion tensor, obtaining that it is identically zero in all cases.

math.DG↗