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Tommaso Sferruzza

Publications and source records attributed to Tommaso Sferruzza.

12 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

Dolbeault formality for complex nilmanifolds

A quasi-isomorphism of differential graded algebras (DGA) is a multiplicative map inducing an isomorphism on cohomology. A DGA is called formal if it can be connected by a chain of quasi-isomorphisms to its cohomology algebra. We prove that the Dolbeault DGA of a complex nilmanifold is formal only if it is a torus, and the Dolbeault algebra of (0,p)-forms is formal if and only if the complex structure is abelian.

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↗
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

Almost complex blow-ups and positive closed $(1,1)$-forms on $4$-dimensional almost complex manifolds

Let $(M,J)$ be a $2n$-dimensional almost complex manifold and let $x\in M$. We define the notion of almost complex blow-up of $(M,J)$ at $x$. We prove the existence of almost complex blow-ups at $x$ under suitable assumptions on the almost complex structure $J$ and we provide explicit examples of such a construction. We note that almost complex blow-ups are unique. When $(M,J)$ is a $4$-dimensional almost complex manifold, we give an obstruction on $J$ to the existence of almost complex blow-ups at a point and prove that the almost complex blow-up at a point of a compact almost Kähler manifold is almost Kähler.

math.DG↗
arXiv2023

Deformations of astheno-Kähler metrics

The property of admitting an astheno-Kähler metric is not stable under the action of small deformations of the complex structure of a compact complex manifold. In this paper, we prove necessary cohomological conditions for the existence of curves of astheno-Kähler metrics along curves of deformations starting from an initial compact complex manifold endowed with an astheno-Kähler metric. Furthermore, we apply our results providing obstructions to the existence of curves of astheno-Kähler metrics on two different families of real $8$-dimensional nilmanifolds endowed with invariant nilpotent complex structures.

math.DG↗
arXiv2022

On cohomological and formal properties of Strong Kähler with torsion and astheno-Kähler metrics

We provide families of compact astheno-Kähler nilmanifolds and we study the behaviour of the complex blowup of such manifolds. We prove that the existence of an astheno-Kähler metric satisfying an extra differential condition is not preserved by blowup. We also study the interplay between Strong Kähler with torsion metrics and geometrically Bott-Chern metrics. We show that Fino-Parton-Salamon nilmanifolds are geometrically-Bott-Chern-formal, whereas we obtain negative results on the product of two copies of primary Kodaira surface, Inoue surface of type $\mathcal{S}_M$ and on the product of a Kodaira surface with an Inoue surface.

math.DG↗
arXiv2022

Deformations of Strong Kähler with torsion metrics

Existence of strong Kähler with torsion metrics, shortly SKT metrics, on complex manifolds has been shown to be unstable under small deformations. We find necessary conditions under which the property of being SKT is stable for a smooth curve of Hermitian metrics $\{ω_t\}_t$ which equals a fixed SKT metric $ω$ for $t=0$, along a differentiable family of complex manifolds $\{M_t\}_t$.

math.DG↗
arXiv2021

$p$-Kähler and balanced structures on nilmanifolds with nilpotent complex structures

Let $(X,J)$ be a nilmanifold with a left-invariant nilpotent complex structure. We study the existence of $p$-Kähler structures (which include Kähler and balanced metrics) on $X$. More precisely, we determine an optimal $p$ such that there are no $p$-Kähler structures on $X$. Finally, we show that, contrarily to the Kähler case, on compact complex manifolds there is no relation between the existence of balanced metrics and the degeneracy step of the Frölicher spectral sequence. More precisely, on balanced manifolds the degeneracy step can be arbitrarily large.

math.DG↗
arXiv2021

Deformations of balanced metrics

Small deformations of the complex structure do not always preserve special metric properties in the Hermitian non-Kähler setting. In this paper, we find necessary conditions for the existence of smooth curves of balanced metrics $\{ω_t\}_t$ which start with a fixed balanced metric $ω$ for $t=0$, along a differentiable family of complex manifolds $\{M_t\}_t$.

math.DG↗
arXiv2019

Geometric formalities along the Chern-Ricci flow

In this paper, we study how the notions of geometric formality according to Kotschick and other geometric formalities adapted to the Hermitian setting evolve under the action of the Chern-Ricci flow on class VII surfaces, including Hopf and Inoue surfaces, and on Kodaira surfaces.

math.DG↗