arXiv research

The article classifies six-dimensional solvmanifolds with non-invariant trivializing sections of their canonical bundle.

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 explic…

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).

Study complex solvmanifolds with trivial canonical bundle and hypercomplex geometry.

Study on Kodaira dimension of specific solvmanifolds without complex structures.

Harmonic almost complex structures on specific Lie groups and solvmanifolds identified.

Study of complex structures on specific solvmanifolds, proving existence and non-existence results.

Study computes invariants on six-dimensional solvmanifolds, providing symplectic structure obstructions.

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 o…

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 precise…

Compute Dolbeault and Bott-Chern cohomologies of complex solvmanifolds.

The paper computes non-trivial triple Massey products on specific non-Kähler solvmanifolds.

New complex non-Kähler manifolds with specific properties are constructed.

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 …

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 $J$ ad…

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{\mathfr…

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 init…

We provide further techniques to study the Dolbeault and Bott-Chern cohomologies of deformations of solvmanifolds by means of finite-dimensional complexes. By these techniques, we can compute the Dolbeault and Bott-Chern cohomologies of some complex solvmanifolds, and we also get explicit examples, showing in particula…

The article confirms a conjecture for solvmanifolds with complex commutator.

New metrics found on complex solvmanifolds.

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 on…

Study Betti and Hodge numbers of solvmanifolds from integer polynomials.

We prove the non-existence of Vaisman metrics on some solvmanifolds with left-invariant complex structures. By this theorem, we show that Oeljeklaus-Toma manifolds does not admit Vaisman metrics.

We prove the non-existence of Vaisman metrics on some solvmanifolds with a left-invariant complex structure. By this theorem, we show that every Oeljeklaus-Toma manifold with $(s,1)$ admits no Vaisman metric.

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 …

Research confirms Streets-Tian conjecture for 2-step solvmanifolds.

Study geometric formal metrics and Massey products on Kähler manifolds with torsion.

Paper connects cohomologies on almost complex manifolds.

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 $G/Γ$, w…

Let $G$ be a simply connected solvable Lie group with a lattice $Γ$ and $N$ the nilradical of $G$. For a complex valued representation $ρ: G\to GL(V_ρ)$ such that the restriction $ρ_{|_{N}}$ is unipotent, as an advanced variation of cohomology computation of solvmanifolds by using Lie algebra cohomology, we construct a…

Study on special metrics and deformations of solvmanifolds.

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…

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.…

Proves conjecture about compatible SKT and balanced metrics on compact solvmanifolds.

New Einstein metric found on non-standard solvmanifold.

In this paper, we study the solvmanifolds constructed from any parabolic subalgebras of any semisimple Lie algebras. These solvmanifolds are naturally homogeneous submanifolds of symmetric spaces of noncompact type. We show that the Ricci curvatures of our solvmanifolds coincide with the restrictions of the Ricci curva…

Study pseudo-Kähler structures on almost abelian solvmanifolds.

We study Bott-Chern cohomology on compact complex non-Kähler surfaces. In particular, we compute such a cohomology for compact complex surfaces in class $\text{VII}$ and for compact complex surfaces diffeomorphic to solvmanifolds.

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 …

New symplectic structures found on complex manifolds without Kähler structures.

The paper explores $p$-Kähler structures on Lie group quotients.

Proves Vaisman solvmanifolds are finite quotients of Kodaira-Thurston manifolds.

The paper constructs pseudo-Iwasawa solvmanifolds with Killing spinors.

Study compact symplectic solvmanifolds' hard Lefschetz property.

We study the Morse-Novikov cohomology and its almost-symplectic counterpart on manifolds admitting locally conformally symplectic structures. More precisely, we introduce lcs cohomologies and we study elliptic Hodge theory, dualities, Hard Lefschetz Condition. We consider solvmanifolds and Oeljeklaus-Toma manifolds. In…

This article is concerned with the study of the holonomy group of flat solvmanifolds. It is known that the holonomy group of a flat solvmanifold is abelian; we give an elementary proof of this fact and moreover we prove that any finite abelian group is the holonomy group of a flat solvmanifold. Furthermore, we show tha…

Characterizes hypercomplex Lie groups and their solvmanifolds.

We obtain new supersymmetric flux vacua of type II supergravities on four-dimensional Minkowski times six-dimensional solvmanifolds. The orientifold O4, O5, O6, O7, or O8-planes and D-branes are localized. All vacua are in addition not T-dual to a vacuum on the torus. The corresponding solvmanifolds are proven to be Ca…