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Adriano Tomassini

Publications and source records attributed to Adriano Tomassini.

At least 19 recordsLinked to original sources
arXiv2026

Non-trivial ABC-Massey products on complex parallelisable solvmanifolds

Triple Aeppli--Bott--Chern--Massey products, shortly, triple ABC-Massey products are higher-order operations on the Bott--Chern and Aeppli cohomologies of a compact complex manifold, and their non-vanishing is an invariant of its pluripotential homotopy type. We prove that every non-Abelian complex unimodular solvable Lie algebra admits a non-vanishing triple ABC--Massey product, and we deduce that so does every compact complex parallelisable solvmanifold.

math.DG↗
arXiv2026

Invariant forms compute the Dolbeault cohomology of complex nilmanifolds

We prove that the inclusion of left-invariant forms into the Dolbeault complex of a compact nilmanifold $M$ endowed with a left-invariant complex structure $J$ induces an isomorphism in cohomology in every bidegree, settling a long-standing conjecture. As consequences, we show that small deformations of $J$ are still invariant, as conjectured by Hasegawa. We also prove that Bott--Chern, Aeppli, and Fr\"olicher invariants are computed by invariant forms and are independent of the lattice, settling a conjecture of Angella on Bott--Chern cohomology.

math.DG↗
arXiv2026

Symplectic non-K\"ahler manifolds with and without the Hard Lefschetz Condition

In this paper we construct compact manifolds without K\"ahler structures that admit both a symplectic form satisfying the Hard Lefschetz Condition (HLC) and another symplectic form that does not. Our construction builds upon the orbifold introduced by Fern\'andez and Mu\~noz and its symplectic resolution studied by Cavalcanti, Fern\'andez, and Mu\~noz. By considering a one-parameter family of symplectic forms on the orbifold, we show that the corresponding resolved manifolds fail to satisfy the HLC for all parameters. However, after performing a suitable symplectic blowup along a union of tori, we obtain a family of symplectic manifolds for which the HLC holds for all non-zero parameters but fails at the central parameter. As a consequence, we exhibit a smooth manifold with no K\"ahler structure whose space of symplectic forms contains both HLC and non-HLC structures in the same connected component. This provides new examples of the subtle interplay between symplectic topology and the Hard Lefschetz property.

math.SG↗
arXiv2026

Holomorphically parallelizable solvmanifolds with special metrics and their deformations

We investigate the existence of strong K\"ahler with torsion metrics along deformations of the Iwasawa manifold and of the holomorphically parallelizable Nakamura manifold. We also show that the class of deformations of the holomorphically parallelizable Nakamura manifold yielding a non-left-invariant complex structure admits a balanced metric but does not admit any strong K\"ahler with torsion metric. We then construct the Kuranishi space of a $4$-dimensional holomorphically parallelizable solvmanifold and study whether small deformations of such a manifold admit SKT metrics. Finally, we provide some results concerning the existence of metrics satisfying $\partial \bar{\partial} \omega = 0$, $\partial \bar{\partial} \omega^2 = 0$ on a particular class of $2$-step nilpotent nilmanifolds.

math.DG↗
arXiv2026

Non-K\"ahler Special Lagrangian submanifolds and SYZ mirror symmetry

We determine purely algebraic equations to identify \textit{SLags} generated by invariant distributions in a class of non-K\"ahler Calabi-Yau manifolds. We determine SLag distributions, determine which leaves integrate to compact submanifolds and study the deformation theory, which we find to be unobstructed. We apply our results to the Iwasawa manifold, the completely solvable 6-dimensional Nakamura manifold and the complex parallelizable Nakamura manifold. Through these examples we find families of topologically distinct \textit{SLags}, including the existence of SLag torus fibrations. Following the proposal of Lau-Tseng-Yau, we compute the non-K\"ahler SYZ mirrors of Nakamura manifolds, together with their refined symplectic Bott-Chern cohomologies. As a consequence, we find the existence of semi-flat non-K\"ahler mirror pairs which are not diffeomorphic.

math.DG↗
arXiv2025

Aeppli-Bott-Chern Massey products on non-K\"ahler solvmanifolds

In this paper, we present explicit computations of non-trivial triple $ABC$-Massey products on non-K\"ahler solvmanifolds endowed with an invariant complex structure. We prove that the {\em Bigalke-Rollenske manifold}, the {\em generalized Nakamura manifolds} satisfying some suitable assumptions and compact quotients of the solvable Lie group $\mathbb{C}^{2n}\ltimes_{\rho} \mathbb{C}^{2m}$ have non-vanishing triple $ABC$-Massey products. Furthermore, such manifolds have no astheno-K\"ahler metric.

math.DG↗
arXiv2025

Kodaira dimension of almost complex $4$-manifolds with torsion first Chern class

In this paper we investigate the Kodaira dimension of almost complex $4$-manifolds with torsion first Chern class. First, we prove that, if the almost complex structure is also tamed, the only possible values for the Kodaira dimension are $0$ or $-\infty$. This is done by developing the theory of pseudoholomorphic structures on vector bundles. In arbitrary dimension, we study infinitesimal deformations of structures with pseudoholomorphically torsion canonical bundle. We compute their tangent space and, under suitable assumptions, we prove an unobstructedness theorem in the spirit of Bogomolov--Tian--Todorov. Together, our results allow to fully describe non-integrable infinitesimal deformations of complex structures on $K3$ and Enriques surfaces in terms of their Kodaira dimension.

math.DG↗
arXiv2025

$\del\delbar$-Lemma and Bott-Chern cohomology of twistor spaces

In the paper we study the Bott-Chern and Aeppli cohomologies of the twistor space of a compact self-dual 4-manifold and we characterize the validity of the $\partial \overline \partial$-lemma. We also compute explicitly the Dolbeault cohomology of the twistor space $Z$ of the flat $4$-dimensional torus, which is known to not satisfy the $\partial\overline{\partial}$ lemma.

math.DG↗
arXiv2025

Hermitian geometrically formal manifolds

We study Hermitian geometrically formal metrics on compact complex manifolds, focusing on Dolbeault, Bott-Chern, and Aeppli cohomologies. We establish topological and cohomological obstructions to their existence and we provide a detailed analysis for compact complex surfaces, complex parallelisable solvmanifolds, and Calabi-Eckmann manifolds. We prove that the standard blow-up metric on any blow-up of a K\"ahler manifold is not geometrically formal, and that K\"ahler metrics with nonnegative curvature operator are necessarily geometrically formal.

math.DG↗
arXiv2025

Hard Lefschetz Condition on symplectic non-K\"ahler solvmanifolds

We provide new families of compact complex manifolds with no K\"ahler structure carrying symplectic structures satisfying the \textit{Hard Lefschetz Condition}. These examples are obtained as compact quotients of the solvable Lie group $\mathbb{C}^{2n} \ltimes_{\rho} \mathbb{C}^{2m}$, for which we construct explicit lattices. By cohomological computations we prove that such manifolds carry symplectic structures satisfying the \textit{Hard Lefschetz Condition}. Furthermore, we compute the Kodaira dimension of an almost-K\"ahler structure and generators for the de Rham and Dolbeault cohomologies.

math.DG↗
arXiv2025

Kodaira dimension of $\mathrm{SU}(m)$-structures

We study the Kodaira dimension of almost complex manifolds admitting an $\mathrm{SU} (m)$-structure. We introduce the notion of almost complex structure of splitting type and of associated $\mathrm{SU} (m)$-structure. When the latter is pseudoholomorphic, we provide two constructions that allow to obtain non-invariant almost complex structures with Kodaira dimension $0$, resp.\ with Kodaira dimension $-\infty$. Our results apply, in particular, to complex structures of splitting type and to several almost complex manifolds already well-studied in the literature

math.DG↗
arXiv2025

Invariant and non-invariant almost complex structures on compact quotients of Lie groups

In this paper we briefly survey the classical problem of understanding which Lie algebras admit a complex structure, put in the broader perspective of almost complex structures with special properties. We focus on the different behavior of invariant and non-invariant structures, with a special attention to their canonical bundle and Kodaira dimension. We provide new examples of computations of Kodaira dimension of invariant and non-invariant structures.

math.DG↗
arXiv2024

$p$-symplectic and $p$-pluriclosed structures on solvmanifolds

Let $(M,J)$ be a $n$-dimensional complex manifold: a $p$-Kähler structure (resp. $p$-symplectic structure) on $M$ is a real, closed $(p,p)$-transverse form $Ω$ (resp. real, closed $2p$-form whose $(p,p)$-component is transverse). We give obstructions to the existence of such structures on compact complex manifolds. We provide several families of compact complex manifolds which admit both $(n-1)$-symplectic structures and special Hermitian metrics.

math.DG↗
arXiv2024

$\partial\bar{\partial}$-Lemma and $p$-Kähler structures on families of solvmanifolds

We provide families of compact $(n + 1)$-dimensional complex non Kähler manifolds satisfying the $\partial\bar{\partial}$-Lemma, with holomoprhically trivial canonical bundle, carrying a balanced metric and with no $p$-Kähler structures. Such a construction extends to the completely solvable case in any dimension Nakamura's construction of low-dimensional holomorphically parallelizable solvmanifolds.

math.DG↗
arXiv2024

Bott-Chern formality and Massey products on strong Kähler with torsion and Kähler solvmanifolds

We study the interplay between geometrically-Bott-Chern-formal metrics and SKT metrics. We prove that a $6$-dimensional nilmanifold endowed with a invariant complex structure admits an SKT metric if and only if it is geometrically-Bott-Chern-formal. We also provide some partial results in higher dimensions for nilmanifolds endowed with a class of suitable complex structures. Furthermore, we prove that any Kähler solvmanifold is geometrically formal. Finally, we explicitly construct lattices for a complex solvable Lie group in the list of Nakamura [23] on which we provide a non vanishing quadruple $ABC$-Massey product.

math.DG↗
arXiv2023

On the spaces of $(d+d^c)$-harmonic forms and $(d+d^\Lambda)$-harmonic forms on almost Hermitian manifolds and complex surfaces

We study the spaces of $(d + d^c)$-harmonic forms and $(d + d^\Lambda)$-harmonic forms, the natural generalization of the spaces of Bott-Chern harmonic forms, resp. symplectic harmonic forms from complex, resp. symplectic, manifolds to almost Hermitian manifolds. With the same techniques, we also prove that Bott-Chern and Aeppli numbers of compact complex surfaces depend only on the topology of the underlying manifold, a fact that was well-known for Hodge numbers of compact complex surfaces. We give several applications to compact quotients of Lie groups by a lattice.

math.DG↗
arXiv2023

Almost complex parallelizable manifolds: Kodaira dimension and special structures

We study the Kodaira dimension of a real parallelizable manifold $M$, with an almost complex structure $J$ in standard form with respect to a given parallelism. For $X = (M, J)$ we give conditions under which $\operatorname{kod}(X) = 0$. We provide examples in the case $M = G \times G$, where $G$ is a compact connected real Lie group. Finally we describe geometrical properties of real parallelizable manifolds in the framework of statistical geometry.

math.DG↗