arXiv subjects

L. Ugarte

Publications and source records attributed to L. Ugarte.

8 recordsLinked to original sources

arXiv2026

On astheno-Kähler nilmanifolds with balanced metrics

In this paper we study the structure of complex nilmanifolds $X$ admitting some special classes of Hermitian metrics, namely, astheno-Kähler, strongly Gauduchon and balanced metrics. We prove that, in complex dimension 4, the existence of a (non necessarily invariant) astheno-Kähler metric on $X$ implies that the nilmanifold is at most $2$-step and it has first Betti number $\geq 6$. Moreover, the complex structure has a very specific form, sometimes called "of special type" in the literature. We also study the interplay between the existence of astheno-Kähler metrics and that of strongly Gauduchon or balanced metrics. A key result is the use of some obstructions that are preserved by what we call $\mathfrak{b}$-extensions. This allows us to study the existence of these metrics on important classes of complex nilmanifolds, such as almost abelian, those having maximal nilpotent complex structures, and 8-dimensional nilmanifolds with non-nilpotent complex structures. We also construct, in every complex dimension $n\geq 4$, complex nilmanifolds admitting both an astheno-Kähler metric (possibly also being strongly Gauduchon) and another metric that is balanced. As an application, astheno-Kähler nilmanifolds with balanced metrics and with Frölicher spectral sequence not degenerating at the second or third pages are found. To our knowledge, these are the first compact astheno-Kähler manifolds with such properties.

math.DG↗

arXiv2025

Nilmanifolds with non-nilpotent complex structures and their pseudo-Kähler geometry

We classify nilpotent Lie algebras with complex structures of weakly non-nilpotent type in real dimension eight, which is the lowest dimension where they arise. Our study, together with previous results on strongly non-nilpotent structures, completes the classification of 8-dimensional nilpotent Lie algebras admitting complex structures of non-nilpotent type. As an application, we identify those that support a pseudo-Kähler metric, thus providing new counterexamples to a previous conjecture and an infinite family of (Ricci-flat) non-flat neutral Calabi-Yau structures. Moreover, we arrive at the topological restriction $b_1(X)\geq 3$ for every pseudo-Kähler nilmanifold $X$ with invariant complex structure, up to complex dimension four.

math.DG↗

arXiv2023

Six dimensional homogeneous spaces with holomorphically trivial canonical bundle

We classify all the $6$-dimensional unimodular Lie algebras $\mathfrak{g}$ admitting a complex structure with non-zero closed $(3,0)$-form. This gives rise to $6$-dimensional compact homogeneous spaces $M=Γ\backslash G$, where $Γ$ is a lattice, admitting an invariant complex structure with holomorphically trivial canonical bundle. As an application, in the balanced Hermitian case, we study the instanton condition for any metric connection $\nabla^{\varepsilon,ρ}$ in the plane generated by the Levi-Civita connection and the Gauduchon line of Hermitian connections. In the setting of the Hull-Strominger system with connection on the tangent bundle being Hermitian-Yang-Mills, we prove that if a compact non-Kähler homogeneous space $M=Γ\backslash G$ admits an invariant solution with respect to some non-flat connection $\nabla$ in the family $\nabla^{\varepsilon,ρ}$, then $M$ is a nilmanifold with underlying Lie algebra $\mathfrak{h}_3$, a solvmanifold with underlying algebra $\mathfrak{g}_7$, or a quotient of the semisimple group SL(2,$\mathbb{C}$). Since it is known that the system can be solved on these spaces, our result implies that they are the unique compact non-Kähler balanced homogeneous spaces admitting such invariant solutions. As another application, on the compact solvmanifold underlying the Nakamura manifold, we construct solutions, on any given balanced Bott-Chern class, to the heterotic equations of motion taking the Chern connection as (flat) instanton.

math.DG↗

arXiv2023

Frölicher spectral sequence of compact complex manifolds with special Hermitian metrics

In this paper we focus on the interplay between the behaviour of the Frölicher spectral sequence and the existence of special Hermitian metrics on the manifold, such as balanced, SKT or generalized Gauduchon. The study of balanced metrics on nilmanifolds endowed with strongly non-nilpotent complex structures allows us to provide infinite families of compact balanced manifolds with Frölicher spectral sequence not degenerating at the second page. Moreover, this result is extended to non-degeneration at any arbitrary page. Similar results are obtained for the Frölicher spectral sequence of compact generalized Gauduchon manifolds. We also find a compact SKT manifold whose Frölicher spectral sequence does not degenerate at the second page, thus providing a counterexample to a conjecture by Popovici.

math.DG↗

arXiv2019

The ascending central series of nilpotent Lie algebras with complex structure

We obtain several restrictions on the terms of the ascending central series of a nilpotent Lie algebra $\mathfrak g$ under the presence of a complex structure $J$. In particular, we find a bound for the dimension of the center of $\mathfrak g$ when it does not contain any non-trivial $J$-invariant ideal. Thanks to these results, we provide a structural theorem describing the ascending central series of 8-dimensional nilpotent Lie algebras $\mathfrak g$ admitting this particular type of complex structures $J$. Since our method is constructive, it allows us to describe the complex structure equations that parametrize all such pairs $(\mathfrak g, J)$.

math.RA↗

arXiv2017

Invariant solutions to the Strominger system and the heterotic equations of motion

We construct many new invariant solutions to the Strominger system with respect to a 2-parameter family of metric connections $\nabla^{\varepsilon,ρ}$ in the anomaly cancellation equation. The ansatz $\nabla^{\varepsilon,ρ}$ is a natural extension of the canonical 1-parameter family of Hermitian connections found by Gauduchon, as one recovers the Chern connection $\nabla^{c}$ for $({\varepsilon,ρ})=(0,\frac12)$, and the Bismut connection $\nabla^{+}$ for $({\varepsilon,ρ})=(\frac12,0)$. In particular, explicit invariant solutions to the Strominger system with respect to the Chern connection, with non-flat instanton and positive $α'$ are obtained. Furthermore, we give invariant solutions to the heterotic equations of motion with respect to the Bismut connection. Our solutions live on three different compact non-Kähler homogeneous spaces, obtained as the quotient by a lattice of maximal rank of a nilpotent Lie group, the semisimple group SL(2,$\mathbb{C}$) and a solvable Lie group. To our knowledge, these are the only known invariant solutions to the heterotic equations of motion, and we conjecture that there is no other such homogeneous space admitting an invariant solution to the heterotic equations of motion with respect to a connection in the ansatz $\nabla^{\varepsilon,ρ}$.

math.DG↗

arXiv2001

Symplectically harmonic cohomology of nilmanifolds

This paper can be considered as an extension to our paper [On symplectically harmonic forms on six-dimensional nilmanifolds, Comment. Math. Helv. 76 (2001), n 1, 89-109]. Also, it contains a brief survey of recent results on symplectically harmonic cohomology.

math.SG↗

arXiv2000

On Symplectically Harmonic Forms on Six-dimensional Nilmanifolds

In the present paper we study the variation of the dimensions $h_k$ of spaces of symplectically harmonic cohomology classes (in the sense of Brylinski) on closed symplectic manifolds. We give a description of such variation for all 6-dimensional nilmanifolds equipped with symplectic forms. In particular, it turns out that certain 6-dimensional nilmanifolds possess families of homogeneous symplectic forms $ω_t$ for which numbers $h_k(M,ω_t)$ vary with respect to t. This gives an affirmative answer to a question raised by Boris Khesin and Dusa McDuff. Our result is in contrast with the case of 4-dimensional nilmanifolds which do not admit such variations by a remark of Dong Yan.

math.SG↗