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Alberto Raffero

Publications and source records attributed to Alberto Raffero.

At least 19 recordsLinked to original sources
arXiv2025

The heterotic G$_2$-system on 2-step nilmanifolds endowed with principal torus bundles

We study the geometric heterotic G$_2$-system on 7-dimensional 2-step nilmanifolds $M=\Gamma\backslash N$ endowed with principal torus bundles and with a prescribed flat tangent bundle instanton. We first prove that every invariant G$_2$-structure solving the system must be coclosed when the dimension of the commutator of $N$ is $1$ or $2$, and under an additional calibration assumption when the dimension is $3$. Then, we discuss the existence of solutions for all possible isomorphism classes of 7-dimensional 2-step nilpotent Lie algebras, and we provide examples with constant dilaton function.

math.DG↗
arXiv2025

Pluripotential theory and special holonomy, I: Hodge decompositions and $\partial\bar\partial$ lemmata

Let $M$ be a compact torsion-free $G_2$ 7-manifold or Calabi-Yau 6-manifold. We prove Hodge decomposition theorems for the $dd^\phi$ operators, introduced by Harvey and Lawson, which generalize the $i\partial\bar\partial$ operator used in classical pluripotential theory. We then obtain analogues of the $\partial\bar\partial$ lemma in this context. We formalize this by defining cohomology spaces analogous to Bott-Chern cohomology and we relate them to harmonic forms on $M$. In the $G_2$ case we provide a geometric interpretation of the corresponding cohomology classes in terms of coassociative submanifolds and gerbes: this is analogous to the classical interpretation of Bott-Chern cohomology classes in terms of divisors and holomorphic line bundles.

math.DG↗
arXiv2024

Three-dimensional positively curved generalized Ricci solitons with SO(3)-symmetries

We prove the existence of a one-parameter family of pairwise non-isometric, complete, positively curved, steady generalized Ricci solitons of gradient type on $\mathbb{R}^3$ that are invariant under the natural cohomogeneity one action of SO(3). In the context of generalized Ricci flow, this result represents the analogue of Bryant's construction of the complete rotationally invariant steady soliton for the Ricci flow.

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arXiv2023

The twisted G$_2$ equation for strong G$_2$-structures with torsion

We discuss general properties of strong G$_2$-structures with torsion and we investigate the twisted G$_2$ equation, which represents the G$_2$-analogue of the twisted Calabi-Yau equation for SU$(n)$-structures introduced by Garcia-Fern\'andez - Rubio - Shahbazi - Tipler. In particular, we show that invariant strong G$_2$-structures with torsion do not occur on compact non-flat solvmanifolds. This implies the non-existence of non-trivial solutions to the twisted Calabi-Yau equation on compact solvmanifolds of dimensions $4$ and $6$. More generally, we prove that a compact, connected homogeneous space admitting invariant strong G$_2$-structures with torsion is diffeomorphic either to $S^3 \times T^4$ or to $S^3 \times S^3 \times S^1$, up to a covering, and that in both cases solutions to the twisted G$_2$ equation exist. Finally, we discuss the behavior of the homogeneous Laplacian coflow for strong G$_2$-structures with torsion on these spaces.

math.DG↗
arXiv2022

Variation formulae for the volume of coassociative submanifolds

We prove new variation formulae for the volume of coassociative submanifolds, expressed in terms of $G_2$ data. As a special case, we obtain a second variation formula for variations within the moduli space of coassociative submanifolds; this formula highlights the role of the ambient torsion and Ricci curvature. These results apply, for example, to coassociative fibrations. We illustrate our formulae with several examples, both homogeneous and non.

math.DG↗
arXiv2022

Infinite families of homogeneous Bismut Ricci flat manifolds

Starting from compact symmetric spaces of inner type, we provide infinite families of compact homogeneous spaces carrying invariant non-flat Bismut connections with vanishing Ricci tensor. These examples turn out to be generalized symmetric spaces of order $4$ and (up to coverings) can be realized as minimal submanifolds of the Bismut flat model spaces, namely compact Lie groups. This construction generalizes the standard Cartan embedding of symmetric spaces.

math.DG↗
arXiv2022

Bismut Ricci flat manifolds with symmetries

We construct examples of compact homogeneous Riemannian manifolds admitting an invariant Bismut connection that is Ricci flat and non-flat, proving in this way that the generalized Alekseevsky-Kimelfeld theorem does not hold. The classification of compact homogeneous Bismut Ricci flat spaces in dimension $5$ is also provided. Moreover, we investigate compact homogeneous spaces with non trivial third Betti number, and we point out other possible ways to construct Bismut Ricci flat manifolds. Finally, since Bismut Ricci flat connections correspond to fixed points of the generalized Ricci flow, we discuss the stability of some of our examples under the flow.

math.DG↗
arXiv2021

Exact G$_2$-structures on compact quotients of Lie groups

We show that the compact quotient $\Gamma\backslash\mathrm{G}$ of a seven-dimensional simply connected Lie group $\mathrm{G}$ by a co-compact discrete subgroup $\Gamma\subset\mathrm{G}$ does not admit any exact $\mathrm{G}_2$-structure which is induced by a left-invariant one on $\mathrm{G}$.

math.DG↗
arXiv2021

Special solutions to the Type IIA flow

We consider the source-free Type IIA flow introduced by Fei-Phong-Picard-Zhang, and we study it in the case where the relevant geometric datum is a symplectic half-flat SU(3)-structure. We show the existence of ancient, immortal and eternal solutions to the flow, provided that the initial symplectic half-flat structure satisfies suitable properties. In particular, we prove that the solution starting at a symplectic half-flat structure with Hermitian Ricci tensor is ancient and evolves self-similarly by scaling the initial datum. These results apply to all known (locally) homogeneous spaces admitting invariant symplectic half-flat SU(3)-structures.

math.DG↗
arXiv2021

Closed G$_2$-structures on unimodular Lie algebras with non-trivial center

We characterize the structure of a seven-dimensional Lie algebra with non-trivial center endowed with a closed G$_2$-structure. Using this result, we classify all unimodular Lie algebras with non-trivial center admitting closed G$_2$-structures, up to isomorphism, and we show that six of them arise as the contactization of a symplectic Lie algebra. Finally, we prove that every semi-algebraic soliton on the contactization of a symplectic Lie algebra must be expanding, and we determine all unimodular Lie algebras with center of dimension at least two that admit semi-algebraic solitons, up to isomorphism.

math.DG↗
arXiv2020

On the dynamical behaviour of the generalized Ricci flow

Motivated by M\"uller-Haslhofer results on the dynamical stability and instability of Ricci-flat metrics under the Ricci flow, we obtain dynamical stability and instability results for pairs of Ricci-flat metrics and vanishing 3-forms under the generalized Ricci flow.

math.DG↗
arXiv2020

Purely coclosed G$_{\mathbf2}$-structures on 2-step nilpotent Lie groups

We consider left-invariant (purely) coclosed G$_2$-structures on 7-dimensional 2-step nilpotent Lie groups. According to the dimension of the commutator subgroup, we obtain various criteria characterizing the Riemannian metrics induced by left-invariant purely coclosed G$_2$-structures. Then, we use them to determine the isomorphism classes of 2-step nilpotent Lie algebras admitting such type of structures. As an intermediate step, we show that every metric on a 2-step nilpotent Lie algebra admitting coclosed G$_2$-structures is induced by one of them. Finally, we use our results to give the explicit description of the metrics induced by purely coclosed G$_2$-structures on 2-step nilpotent Lie algebras with derived algebra of dimension at most two, up to automorphism.

math.DG↗
arXiv2020

Recent results on closed G$_2$-structures

We review recent results concerning closed G$_2$-structures on seven-dimensional manifolds. In particular, we discuss the construction of examples and some related problems.

math.DG↗
arXiv2020

Exact $G_2$-structures on unimodular Lie algebras

We consider seven-dimensional unimodular Lie algebras $\mathfrak{g}$ admitting exact $G_2$-structures, focusing our attention on those with vanishing third Betti number $b_3(\mathfrak{g})$. We discuss some examples, both in the case when $b_2(\mathfrak{g})\neq0$, and in the case when the Lie algebra $\mathfrak{g}$ is (2,3)-trivial, i.e., when both $b_2(\mathfrak{g})$ and $b_3(\mathfrak{g})$ vanish. These examples are solvable, as $b_3(\mathfrak{g})=0$, but they are not strongly unimodular, a necessary condition for the existence of lattices on the simply connected Lie group corresponding to $\mathfrak{g}$. More generally, we prove that any seven-dimensional (2,3)-trivial strongly unimodular Lie algebra does not admit any exact $G_2$-structure. From this, it follows that there are no compact examples of the form $(Γ\backslash G,φ)$, where $G$ is a seven-dimensional simply connected Lie group with (2,3)-trivial Lie algebra, $Γ\subset G$ is a co-compact discrete subgroup, and $φ$ is an exact $G_2$-structure on $Γ\backslash G$ induced by a left-invariant one on $G$.

math.DG↗
arXiv2019

Closed G$_2$-structures with a transitive reductive group of automorphisms

We provide the complete classification of seven-dimensional manifolds endowed with a closed non-parallel G$_2$-structure and admitting a transitive reductive group G of automorphisms. In particular, we show that the center of G is one-dimensional and the manifold is the Riemannian product of a flat factor and a non-compact homogeneous six-dimensional manifold endowed with an invariant strictly symplectic half-flat SU(3)-structure.

math.DG↗
arXiv2019

Homogeneous 8-manifolds admitting invariant Spin(7)-structures

We study compact, simply connected, homogeneous 8-manifolds admitting invariant Spin(7)-structures, classifying all canonical presentations G/H of such spaces, with G simply connected. For each presentation, we exhibit explicit examples of invariant Spin(7)-structures and we describe their type, according to Fern\'andez classification. Finally, we analyse the associated Spin(7)-connection with torsion.

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