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Anna Fino

Publications and source records attributed to Anna Fino.

At least 73 records · Page 4Linked to original sources
arXiv2017

Coclosed $G_2$-structures inducing nilsolitons

We show obstructions to the existence of a coclosed $G_2$-structure on a Lie algebra $\mathfrak g$ of dimension seven with non-trivial center. In particular, we prove that if there exist a Lie algebra epimorphism from $\mathfrak g$ to a six-dimensional Lie algebra $\mathfrak h$, with kernel contained in the center of $\mathfrak g$, then any coclosed $G_2$-structure on $\mathfrak g$ induces a closed and stable three form on $\mathfrak h$ that defines an almost complex structure on $\mathfrak h$. As a consequence, we obtain a classification of the 2-step nilpotent Lie algebras which carry coclosed $G_2$-structures. We also prove that each one of these Lie algebras has a coclosed $G_2$-structure inducing a nilsoliton metric, but this is not true for 3-step nilpotent Lie algebras with coclosed $G_2$-structures. The existence of contact metric structures is also studied.

math.DG↗
arXiv2016

On de Rham and Dolbeault Cohomology of Solvmanifolds

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 $\tilde{\mathfrak g}$ whose cohomology is isomorphic to the de Rham cohomology of $G/Γ$ by using a modification of $G$ associated with a algebraic sub-torus of the Zariski-closure of the image of the adjoint representation. This technique includes the construction due to Guan and developed by the first two authors. In this paper, we also give a Dolbeault version of such technique for complex solvmanifolds, i.e. for solvmanifolds endowed with an invariant complex structure. We construct a finite dimensional cochain complex which computes the Dolbeault cohomology of a complex solvmanifold $G/Γ$ with holomorphic Mostow bundle and we give a construction of a new Lie algebra $\breve {\mathfrak g}$ with a complex structure whose cohomology is isomorphic to the Dolbeault cohomology of $G/Γ$.

math.DG↗
arXiv2016

Algebraic dimension of complex nilmanifolds

Let M be a complex nilmanifold, that is, a compact quotient of a nilpotent Lie group endowed with an invariant complex structure by a discrete lattice. A holomorphic differential on M is a closed, holomorphic 1-form. We show that $a(M)\leq k$, where $a(M)$ is the algebraic dimension $a(M)$ (i.e. the transcendence degree of the field of meromorphic functions) and $k$ is the dimension of the space of holomorphic differentials. We prove a similar result about meromorphic maps to Kahler manifolds.

math.DG↗
arXiv2016

Holonomy groups of $G_2^*$-manifolds

We classify the holonomy algebras of manifolds admitting an indecomposable torsion free $G_2^*$-structure, i.e. for which the holonomy representation does not leave invariant any proper non-degenerate subspace. We realize some of these Lie algebras as holonomy algebras of left-invariant metrics on Lie groups.

math.DG↗
arXiv2016

Holomorphic Poisson Structures and its Cohomology on Nilmanifolds

The subject for investigation in this note is concerned with holomorphic Poisson structures on nilmanifolds with abelian complex structures. As a basic fact, we establish that on such manifolds, the Dolbeault cohomology with coefficients in holomorphic polyvector fields is isomorphic to the cohomology of invariant forms with coefficients in invariant polyvector fields. We then quickly identify the existence of invariant holomorphic Poisson structures. More important, the spectral sequence of the Poisson bi-complex associated to such holomorphic Poisson structure degenerates at $E_2$. We will also provide examples of holomorphic Poisson structures on such manifolds so that the related spectral sequence does not degenerate at $E_2$.

math.DG↗
arXiv2015

Locally conformal calibrated $G_2$-manifolds

We study conditions for which the mapping torus of a 6-manifold endowed with an $SU(3)$-structure is a locally conformal calibrated $G_2$-manifold, that is, a 7-manifold endowed with a $G_2$-structure $φ$ such that $d φ= - θ\wedge φ$ for a closed non-vanishing 1-form $θ$. Moreover, we show that if $(M, φ)$ is a compact locally conformal calibrated $G_2$-manifold with $\mathcal{L}_{θ^{\#}} φ=0$, where ${θ^{\#}}$ is the dual of $θ$ with respect to the Riemannian metric $g_φ$ induced by $φ$, then $M$ is a fiber bundle over $S^1$ with a coupled $SU(3)$-manifold as fiber.

math.DG↗
arXiv2015

On the existence of balanced and SKT metrics on nilmanifolds

On a complex manifold an Hermitian metric which is simultaneously SKT and balanced has to be necessarily Kähler. It has been conjectured that if a compact complex manifold (M,J) has an SKT metric and a balanced metric both compatible with J, then (M, J) is necessarily Kähler. We show that the conjecture is true for nilmanifolds.

math.DG↗
arXiv2015

Tamed symplectic structures on compact solvmanifolds of completely solvable type

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 $J$ admits a symplectic form taming J if and only if $M$ is a complex torus. This result generalizes the one obtained in [7] for nilmanifolds.

math.DG↗
arXiv2015

Coupled SU(3)-structures and Supersymmetry

We review coupled ${\rm SU}(3)$-structures, also known in the literature as restricted half-flat structures, in relation to supersymmetry. In particular, we study special classes of examples admitting such structures and the behaviour of flows of ${\rm SU}(3)$-structures with respect to the coupled condition.

math.DG↗
arXiv2015

Einstein locally conformal calibrated $G_2$-structures

We study locally conformal calibrated $G_2$-structures whose underlying Riemannian metric is Einstein, showing that in the compact case the scalar curvature cannot be positive. As a consequence, a compact homogeneous $7$-manifold cannot admit an invariant Einstein locally conformal calibrated $G_2$-structure unless the underlying metric is flat. In contrast to the compact case, we provide a non-compact example of homogeneous manifold endowed with a locally conformal calibrated $G_2$-structure whose associated Riemannian metric is Einstein and non Ricci-flat. The homogeneous Einstein metric is a rank-one extension of a Ricci soliton on the $3$-dimensional complex Heisenberg group endowed with a left-invariant coupled ${\rm SU}(3)$-structure $(ω, Ψ)$, i.e., such that $d ω= c {\rm Re}(Ψ)$, with $c \in \mathbb{R} - \{ 0 \}$. Nilpotent Lie algebras admitting a coupled ${\rm SU}(3)$-structure are also classified.

math.DG↗
arXiv2015

Six dimensional solvmanifolds with holomorphically trivial canonical bundle

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. As an application we construct a holomorphic family $(M,J_a)$ of compact complex manifolds such that $(M,J_a)$ satisfies the $\partial\bar\partial$-Lemma and admits a balanced metric for any $a\not=0$, but the central limit neither satisfies the $\partial\bar\partial$-Lemma nor admits balanced metrics.

math.DG↗
arXiv2015

Laplacian flow of closed $G_2$-structures inducing nilsolitons

We study the existence of left invariant closed $G_2$-structures defining a Ricci soliton metric on simply connected nonabelian nilpotent Lie groups. For each one of these $G_2$-structures, we show long time existence and uniqueness of solution for the Laplacian flow on the noncompact manifold. Moreover, considering the Laplacian flow on the associated Lie algebra as a bracket flow on $\R^7$ in a similar way as in [23] we prove that the underlying metrics $g(t)$ of the solution converge smoothly, up to pull-back by time-dependent diffeomorphisms, to a flat metric, uniformly on compact sets in the nilpotent Lie group, as $t$ goes to infinity.

math.DG↗
arXiv2015

Special Hermitian metrics on compact solvmanifolds

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 initial datum. Moreover, we state a new conjecture about the existence of balanced and SKT metrics on compact complex manifolds. We show that the conjecture is true for nilmanifolds of dimension 6 and 8 and for 6-dimensional solvmanifolds with holomorphically trivial canonical bundle.

math.DG↗
arXiv2015

Four-dimensional pseudo-Riemannian homogeneous Ricci solitons

We consider four-dimensional homogeneous pseudo-Riemannian manifolds with non-trivial isotropy and completely classify the cases giving rise to non-trivial homogeneous Ricci solitons. In particular, we show the existence of non-compact homogeneous (and also invariant) pseudo-Riemannian Ricci solitons which are not isometric to solvmanifolds, and of conformally flat homogeneous pseudo-Riemannian Ricci solitons which are not symmetric.

math.DG↗
arXiv2014

SKT and Tamed Symplectic structures on solvmanifolds

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 of the nilradical of $G$ or $J$ is abelian or $G$ is almost abelian (not of type (I)), then the solvmanifold $G/Γ$ cannot admit any symplectic form taming the complex structure $J$, unless $G/Γ$ is Kähler. As a consequence, we show that the family of non-Kähler complex manifolds constructed by Oeljeklaus and Toma cannot admit any symplectic form taming the complex structure.

math.DG↗
arXiv2014

Torsion-free $G_{2(2)}^*$-structures with full holonomy on nilmanifolds

We study the existence of invariant metrics with holonomy $G_{2(2)}^* \subset SO(4,3)$ on compact nilmanifolds, i.e. on compact quotients of nilpotent Lie groups by discrete subgroups. We prove that, up to isomorphism, there exists only one indecomposable nilpotent Lie algebra admitting a torsion-free $G_{2(2)}^*$-structure such that the center is definite with respect to the induced inner product. In particular, we show that the associated compact nilmanifold admits a 3-parameter family of invariant metrics with full holonomy $G_{2(2)}^*$.

math.DG↗
arXiv2013

G_2-structures on Einstein solvmanifolds

We study the $G_2$ analogue of the Goldberg conjecture on non-compact solvmanifolds. In contrast to the almost-Kähler case we prove that a 7-dimensional solvmanifold cannot admit any left-invariant calibrated $G_2$-structure $φ$ such that the induced metric $g_φ$ is Einstein, unless $g_φ$ is flat. We give an example of 7-dimensional solvmanifold admitting a left-invariant calibrated $G_2$-structure $φ$ such that $g_φ$ is Ricci-soliton. Moreover, we show that a 7-dimensional (non-flat) Einstein solvmanifold $(S,g)$ cannot admit any left-invariant cocalibrated $G_2$-structure $φ$ such that the induced metric $g_φ = g$.

math.DG↗