arXiv subjects
Publications and source records attributed to Anna Fino.
Starting from the product of a $3$-torus and a compact Kähler (respectively, hyperKähler) manifold we construct via mapping tori generalized Kähler manifolds of split (respectively, non-split) type. In this way we obtain new non-Kähler examples and we recover the known examples of generalized Kähler solvmanifolds. Moreover, we investigate the formality and the Dolbeault cohomology of the generalized Kähler mapping tori.
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-Fernández - 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.
A $p$-Kähler structure on a complex manifold of complex dimension $n$ is given by a $d$-closed transverse real $(p,p)$-form. In the paper we study the existence of $p$-Kähler structures on compact quotients of simply connected Lie groups by discrete subgroups endowed with an invariant complex structure. In particular, we discuss the existence of $p$-Kähler structures on nilmanifolds, with a focus on the case $p =2$ and complex dimension $n = 4$. Moreover, we prove that a $(n-2)$-Kähler almost abelian solvmanifold of complex dimension $n\geq3$ has to be Kähler.
In this note, we prove the existence of homogeneous gradient solitons for the G$_2$-Laplacian flow by providing the first known example of this type. This result singles out the G$_2$-Laplacian flow as the first known geometric flow admitting homogeneous gradient solitons on spaces that are one-dimensional extensions.
In this note we study $T^2$-invariant pluriclosed metrics on the Kodaira-Thurston surface. We obtain a characterization of $T^2$-invariant Vaisman metrics, and notice that the Kodaira-Thurston surface admits Vaisman metrics with non-constant scalar curvature. Then we study the behaviour of the Vaisman condition in relation to the pluriclosed flow. As a consequence, we show that if the initial metric on the Kodaira-Thurston surface is a $T^2$-invariant Vaisman metric, then the pluriclosed flow preserves the Vaisman condition, extending to the non-constant scalar curvature case the previous results in [6].
We complete the classification of six-dimensional strongly unimodular almost nilpotent Lie algebras admitting complex structures. For several cases we describe the space of complex structures up to isomorphism. As a consequence we determine the six-dimensional almost nilpotent solvmanifolds admitting an invariant complex structure and study the existence of special types of Hermitian metrics, including SKT, balanced, locally conformally Kähler, and strongly Gauduchon metrics. In particular, we determine new balanced solvmanifolds and confirm a conjecture by the first author and Vezzoni regarding SKT and balanced structures in the six-dimensional strongly unimodular almost nilpotent case. Moreover, we prove some negative results regarding complex structures tamed by symplectic forms, showing in particular that in every dimension such structures cannot exist on non-Kähler almost abelian Lie algebras.
A Hermitian metric on a complex manifold $(M, I)$ of complex dimension $n$ is called Calabi-Yau with torsion (CYT) or Bismut-Ricci flat, if the restricted holonomy of the associated Bismut connection is contained in ${\rm SU}(n)$ and it is called strong Kähler with torsion (SKT) or pluriclosed if the associated fundamental form $F$ is $\partial \overline \partial$-closed. In the paper we study the existence of left-invariant SKT and CYT metrics on compact semi-simple Lie groups endowed with a Samelson complex structure $I$. In particular, we show that if $I$ is determined by some maximal torus $T$ and $g$ is a left-invariant Hermitian metric, which is also invariant under the right action of the torus $T$, and is both CYT and SKT, then $g$ has to be Bismut flat.
We construct new examples of non-formal simply connected compact Sasaki-Einstein 7-manifolds. We determine the minimal model of the total space of any fibre bundle over $CP^2$ with fibre $S^1\times S^2$ or $S^3/Z_p$ ($p>0$), and we apply this to conclude that the Aloff-Wallach spaces are formal. We also find examples of formal manifolds and non-formal manifolds, which are locally conformal parallel $Spin(7)$-manifolds. On the other hand, we construct associative minimal submanifolds in the Aloff-Wallach spaces and in any regular Sasaki-Einstein 7-manifold; in particular, in the space $Q(1,1,1)=(SU(2) \times SU(2) \times SU(2))/ (U(1) \times U(1))$ with the natural $S^1$-family of nearly parallel $G_2$-structures induced by the Sasaki-Einstein structure. In each of those cases, we obtain a family of non-trivial associative deformations.
We investigate the geometry of almost Robinson manifolds, Lorentzian analogues of almost Hermitian manifolds, defined by Nurowski and Trautman as Lorentzian manifolds of even dimension equipped with a totally null complex distribution of maximal rank. Associated to such a structure, there is a congruence of null curves, which, in dimension four, is geodesic and non-shearing if and only if the complex distribution is involutive. Under suitable conditions, the distribution gives rise to an almost Cauchy-Riemann structure on the leaf space of the congruence. We give a comprehensive classification of such manifolds on the basis of their intrinsic torsion. This includes an investigation of the relation between an almost Robinson structure and the geometric properties of the leaf space of its congruence. We also obtain conformally invariant properties of such a structure, and we finally study an analogue of so-called generalised optical geometries as introduced by Robinson and Trautman.
We study compact almost complex manifolds admitting a Hermitian metric satisfying an integral condition involving $\overline \partial$-harmonic $(0,1)$-forms. We prove that this integral condition is automatically satisfied, if the Hermitian metric on the compact almost complex manifold is strongly Gauduchon. Under the further assumption that the almost complex structure is integrable, we show that the integral condition for a Gauduchon metric is equivalent to be strongly Gauduchon. In particular, a compact complex surface with a Gauduchon metric satisfying the integral condition is automatically Kähler. If we drop the integrability assumption on the complex structure, we show that there exists a compact almost complex $4$-dimensional manifold with a Hermitian metric satisfying the integral condition, but which does not admit any compatible almost-Kähler metric.
A locally conformal SKT (shortly LCSKT) structure is a Hermitian structure $(J, g)$ whose Bismut torsion 3-form $H$ satisfies the condition $dH = α\wedge H$, for some closed non-zero 1-form $α$. This condition was introduced in [6] as a generalization of the SKT (or pluriclosed) condition $dH= 0$. In this paper, we characterize the almost abelian Lie algebras admitting a Hermitian structure $(J, g)$ such that $dH = α\wedge H$, for some closed 1-form $α$. As an application we classifiy LCSKT almost abelian Lie algebras in dimension $6$. Finally, we also study on almost abelian Lie algebras the compatibility between the LCSKT condition and other types of Hermitian structures.
A Hermitian metric on a complex manifold is called SKT (strong Kähler with torsion) if the Bismut torsion $3$-form $H$ is closed. As the conformal generalization of the SKT condition, we introduce a new type of Hermitian structure, called \emph{locally conformal SKT} (or shortly LCSKT). More precisely, a Hermitian structure $(J,g)$ is said to be LCSKT if there exists a closed non-zero $1$-form $α$ such that $d H = α\wedge H$. In the paper we consider non-trivial LCSKT structures, i.e. we assume that $d H \neq 0$ and we study their existence on Lie groups and their compact quotients by lattices. In particular, we classify 6-dimensional nilpotent Lie algebras admitting a LCSKT structure and we show that, in contrast to the SKT case, there exists a $6$-dimensional $3$-step nilpotent Lie algebra admitting a non-trivial LCSKT structure. Moreover, we show a characterization of even dimensional almost abelian Lie algebras admitting a non-trivial LCSKT structure, which allows us to construct explicit examples of $6$-dimensional unimodular almost abelian Lie algebras admitting a non-trivial LCSKT structure. The compatibility between the LCSKT and the balanced condition is also discussed, showing that a Hermitian structure on a 6-dimensional nilpotent or a $2n$-dimensional almost abelian Lie algebra cannot be simultaneously LCSKT and balanced, unless it is Kähler.
We construct a compact formal 7-manifold with a closed $G_2$-structure and with first Betti number $b_1=1$, which does not admit any torsion-free $G_2$-structure, that is, it does not admit any $G_2$-structure such that the holonomy group of the associated metric is a subgroup of $G_2$. We also construct associative calibrated (hence volume-minimizing) 3-tori with respect to this closed $G_2$-structure and, for each of those 3-tori, we show a 3-dimensional family of non-trivial associative deformations. We also construct a fibration of our 7-manifold over $S^2\times S^1$ with generic fiber a (non-calibrated) coassociative 4-torus and some singular fibers.
We show that the compact quotient $Γ\backslash\mathrm{G}$ of a seven-dimensional simply connected Lie group $\mathrm{G}$ by a co-compact discrete subgroup $Γ\subset\mathrm{G}$ does not admit any exact $\mathrm{G}_2$-structure which is induced by a left-invariant one on $\mathrm{G}$.
In this paper we investigate the existence of invariant SKT, balanced and generalized Kähler structures on compact quotients $Γ\backslash G$, where $G$ is an almost nilpotent Lie group whose nilradical has one-dimensional commutator and $Γ$ is a lattice of $G$. We first obtain a characterization of Hermitian almost nilpotent Lie algebras $\mathfrak{g}$ whose nilradical $\mathfrak{n}$ has one-dimensional commutator and a classification result in real dimension six. Then, we study the ones admitting SKT and balanced structures and we examine the behaviour of such structures under flows. In particular, we construct new examples of compact SKT manifolds. Finally, we prove some non-existence results for generalized Kähler structures in real dimension six. In higher dimension we construct the first examples of non-split generalized Kähler structures (i.e., such that the associated complex structures do not commute) on almost abelian Lie algebras. This leads to new compact (non-Kähler) manifolds admitting non-split generalized Kähler structures.
We study balanced Hermitian structures on almost abelian Lie algebras, i.e. on Lie algebras with a codimension-one abelian ideal. In particular, we classify six-dimensional almost abelian Lie algebras which carry a balanced structure. It has been conjectured by A. Fino and L. Vezzoni that a compact complex manifold admitting both a balanced metric and a SKT metric necessarily has a Kähler metric: we prove this conjecture for compact almost abelian solvmanifolds with left-invariant complex structures. Moreover, we investigate the behaviour of the flow of balanced metrics introduced by L. Bedulli and L. Vezzoni and of the anomaly flow by D. H. Phong, S. Picard and X. Zhang on almost abelian Lie groups. In particular, we show that the anomaly flow preserves the balanced condition and that locally conformally Kähler metrics are fixed points.
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.
We construct new smooth solutions to the Hull-Strominger system, showing that the Fu-Yau solution on torus bundles over K3 surfaces can be generalized to torus bundles over K3 orbifolds. In particular, we prove that, for $13 \leq k \leq 22$ and $14\leq r\leq 22$, the smooth manifolds $S^1\times \sharp_k(S^2\times S^3)$ and $\sharp_r (S^2 \times S^4) \sharp_{r+1} (S^3 \times S^3)$, have a complex structure with trivial canonical bundle and admit a solution to the Hull-Strominger system.