arXiv subjects

Anna Fino

Publications and source records attributed to Anna Fino.

At least 91 records · Page 5Linked to original sources

arXiv2013

The pluriclosed flow on nilmanifolds and Tamed symplectic forms

We study evolution of (strong Kähler with torsion) SKT structures via the pluriclosed flow on complex nilmanifolds, i.e. on compact quotients of simply connected nilpotent Lie groups by discrete subgroups endowed with an invariant complex structure. Adapting to our case the techniques introduced by Jorge Lauret for studying Ricci flow on homogeneous spaces we show that for SKT Lie algebras the pluriclosed flow is equivalent to a bracket flow and we prove a long time existence result in the nilpotent case. Finally, we introduce a natural flow for evolving tamed symplectic forms on a complex manifold, by considering evolution of symplectic forms via the flow induced by the Bismut Ricci form.

math.DG↗

arXiv2012

The Calabi-Yau equation for $T^2$-bundles over $\mathbb{T}^2$: the non-Lagrangian case

In the spirit of [10,2], we study the Calabi-Yau equation on $T^2$-bundles over $\mathbb{T}^2$ endowed with an invariant non-Lagrangian almost-Kähler structure showing that for $T^2$-invariant initial data it reduces to a Monge-Ampère equation having a unique solution. In this way we prove that for every total space $M^4$ of an orientable $T^2$-bundle over $\mathbb{T}^2$ endowed with an invariant almost-Kähler structure the Calabi-Yau problem has a solution for every normalized $T^2$-invariant volume form.

math.DG↗

arXiv2012

On generalized Gauduchon metrics

We study a class of Hermitian metrics on complex manifolds, recently introduced by J. Fu, Z. Wang and D. Wu, which are a generalization of Gauduchon metrics. This class includes the one of Hermitian metrics for which the associated fundamental 2-form is $\partial \bar\partial$-closed. Examples are given on nilmanifolds, on products of Sasakian manifolds, on $S^1$-bundles and via the twist construction introduced by A. Swann.

math.DG↗

arXiv2011

Tamed Symplectic forms and Generalized Geometry

We show that symplectic forms taming complex structures on compact manifolds are related to special types of almost generalized Kähler structures. By considering the commutator $Q$ of the two associated almost complex structures $J_{\pm}$, we prove that if either the manifold is 4-dimensional or the distribution ${Im} \, Q$ is involutive, then the manifold can be expressed locally as a disjoint union of twisted Poisson leaves.

math.DG↗

arXiv2011

Analog of selfduality in dimension nine

We introduce a type of Riemannian geometry in nine dimensions, which can be viewed as the counterpart of selfduality in four dimensions. This geometry is related to a 9-dimensional irreducible representation of ${\bf SO}(3) \times {\bf SO} (3)$ and it turns out to be defined by a differential 4-form. Structures admitting a metric connection with totally antisymmetric torsion and preserving the 4-form are studied in detail, producing locally homogeneous examples which can be viewed as analogs of self-dual 4-manifolds in dimension nine.

math.DG↗

arXiv2011

Special Hermitian metrics and Lie groups

A Hermitian metric on a complex manifold is called strong Kähler with torsion (SKT) if its fundamental 2-form $ω$ is $\partial \bar \partial$-closed. We review some properties of strong KT metrics also in relation with symplectic forms taming complex structures. Starting from a $2n$-dimensional SKT Lie algebra $\mathfrak g$ {and using} a Hermitian flat connection on $\mathfrak g$ we construct a $4n$-dimensional SKT Lie algebra. We apply this method to some 4-dimensional SKT Lie algebras. Moreover, we classify symplectic forms taming complex structures on 4-dimensional Lie algebras.

math.DG↗

arXiv2011

The Calabi-Yau equation on 4-manifolds over 2-tori

This paper pursues the study of the Calabi-Yau equation on certain symplectic non-Kaehler 4-manifolds, building on a key example of Tosatti-Weinkove in which more general theory had proved less effective. Symplectic 4-manifolds admitting a 2-torus fibration over a 2-torus base are modelled on one of three solvable Lie groups. Having assigned an invariant almost-Kaehler structure and a volume form that effectively varies only on the base, one seeks a symplectic form with this volume. Our approach simplifies the previous analysis of the problem, and establishes the existence of solutions in various other cases.

math.DG↗

arXiv2011

On the geometry underlying supersymmetric flux vacua with intermediate SU(2) structure

We show that supersymmetric flux vacua with intermediate SU(2) structure is closely related to some special classes of half-flat structures. More concretely, solutions of the SUSY equations IIA possess a symplectic half-flat structure, whereas solutions of the SUSY equations IIB admit a half-flat structure which is in certain sense near to the balanced condition. Using this result we show that compact simply connected manifolds do not admit type IIB solutions. New solutions of the SUSY equations IIA and IIB are constructed from hyperkähler 4-manifolds, special hypo 5-manifolds and 6-dimensional solvmanifolds.

math.DG↗

arXiv2011

Tamed Symplectic forms and SKT metrics

Symplectic forms taming complex structures on compact manifolds are strictly related to Hermitian metrics having the fundamental form $\partial \bar \partial $-closed, i.e. to strong Kähler with torsion (${\rm SKT}$) metrics. It is still an open problem to exhibit a compact example of a complex manifold having a tamed symplectic structure but non-admitting Kähler structures. We show some negative results for the existence of symplectic forms taming complex structures on compact quotients of Lie groups by discrete subgroups. In particular, we prove that if $M$ is a nilmanifold (not a torus) endowed with an invariant complex structure $J$, then $(M, J)$ does not admit any symplectic form taming $J$. Moreover, we show that if a nilmanifold $M$ endowed with an invariant complex structure $J$ admits an ${\rm SKT}$ metric, then $M$ is at most 2-step. As a consequence we classify 8-dimensional nilmanifolds endowed with an invariant complex structure admitting an SKT metric.

math.DG↗

arXiv2010

Strong Kaehler with torsion structures from almost contact manifolds

For an almost contact metric manifold $N$, we find conditions for which either the total space of an $S^1$-bundle over $N$ or the Riemannian cone over $N$ admits a strong Kähler with torsion (SKT) structure. In this way we construct new 6-dimensional SKT manifolds. Moreover, we study the geometric structure induced on a hypersurface of an SKT manifold, and use such structures to construct new SKT manifolds via appropriate evolution equations. Hyper-Kähler with torsion (HKT) structures on the total space of an $S^1$-bundle over manifolds with three almost contact structures are also studied.

math.DG↗

arXiv2010

On astheno-Kaehler metrics

A Hermitian metric on a complex manifold of complex dimension $n$ is called {\em astheno-Kähler} if its fundamental $2$-form $F$ satisfies the condition $\partial \overline \partial F^{n - 2} =0$. If $n =3$, then the metric is {\em strong KT}, i.e. $F$ is $\partial \overline \partial$-closed. By using blow-ups and the twist construction, we construct simply-connected astheno-Kähler manifolds of complex dimension $n > 3$. Moreover, we construct a family of astheno-Kähler (non strong KT) $2$-step nilmanifolds of complex dimension $4$ and we study deformations of strong KT structures on nilmanifolds of complex dimension $3$. Finally, we study the relation between astheno-Kähler condition and (locally) conformally balanced one and we provide examples of locally conformally balanced astheno-Kähler metrics on $\T^2$-bundles over (non-Kähler) homogeneous complex surfaces.

math.DG↗

arXiv2010

Some results on cosymplectic manifolds

We obtain a generalization of the Kodaira-Morrow stability theorem for cosymplectic structures. We investigate cosymplectic geometry on Lie groups and on their compact quotients by uniform discrete subgroups. In this way we show that a compact solvmanifold admits a cosymplectic structure if and only if it is a finite quotient of a torus.

math.DG↗

arXiv2010

Calabi-Yau cones from contact reduction

We consider a generalization of Einstein-Sasaki manifolds, which we characterize in terms both of spinors and differential forms, that in the real analytic case corresponds to contact manifolds whose symplectic cone is Calabi-Yau. We construct solvable examples in seven dimensions. Then, we consider circle actions that preserve the structure, and determine conditions for the contact reduction to carry an induced structure of the same type. We apply this construction to obtain a new hypo-contact structure on S^2\times T^3.

math.DG↗

arXiv2009

On the de Rham cohomology of solvmanifolds

By using results by D. Witte on the superigidity of lattices in solvable Lie groups we get a different proof of a recent remarkable result obtained by D. Guan on the de Rham cohomology of a compact solvmanifold, i.e. of a quotient of a connected and simply connected solvable Lie group $G$ by a lattice $Γ$. This result can be applied to compute the Betti numbers of a compact solvmanifold $G/Γ$ even in the case that the solvable Lie group $G$ and the lattice $Γ$ do not satisfy the Mostow condition.

math.DG↗

arXiv2009

On some cohomological properties of almost complex manifolds

We study a special type of almost complex structures, called pure and full and introduced by T.J. Li and W. Zhang, in relation to symplectic structures and Hard Lefschetz condition. We provide sufficient conditions to the existence of the above type of almost complex structures on compact quotients of Lie groups by discrete subgroups. We obtain families of pure and full almost complex structures on compact nilmanifolds and solvmanifolds. Some of these families are parametrized by real 2-forms which are anti-invariant with respect to the almost complex structures.

math.DG↗

arXiv2009

A class of Sasakian 5-manifolds

We obtain some general results on Sasakian Lie algebras and prove as a consequence that a (2n + 1)-dimensional nilpotent Lie group admitting left-invariant Sasakian structures is isomorphic to the real Heisenberg group $H_{2n + 1}$. Furthermore, we classify Sasakian Lie algebras of dimension 5 and determine which of them carry a Sasakian $α$-Einstein structure. We show that a 5-dimensional solvable Lie group with a left-invariant Sasakian structure and which admits a compact quotient by a discrete subgroup is isomorphic to either $H_5$ or a semidirect product $\R \ltimes (H_3 \times \R)$. In particular, the compact quotient is an $S^1$-bundle over a 4-dimensional Kähler solvmanifold.

math.DG↗

arXiv2009

Blow-ups and resolutions of strong Kähler with torsion metrics

On a compact complex manifold we study the behaviour of strong Kähler with torsion (strong KT) structures under small deformations of the complex structure and the problem of extension of a strong KT metric. In this context we obtain the analogous result of Miyaoka extension theorem. Studying the blow-up of a strong KT manifold at a point or along a complex submanifold, we prove that a complex orbifold endowed with a strong KT metric admits a strong KT resolution. In this way we obtain new examples of compact simply-connected strong KT manifolds.

math.DG↗

arXiv2008

Non Kaehler solvmanifolds with generalized Kaehler structure

We construct a compact 6-dimensional solvmanifold endowed with a non-trivial invariant generalized Kähler structure and which does not admit any Kähler metric. This is in contrast with the case of nilmanifolds which cannot admit any invariant generalized Kähler structure unless they are tori.

math.DG↗