Symplectic forms on six dimensional real solvable Lie algebras I

Special symplectic six-manifolds

We classify nilmanifolds with an invariant symplectic half-flat structure. We solve the half-flat evolution equations in one example, writing down the resulting Ricci-flat metric. We study the geometry of the orbit space of 6-manifolds with an SU(3)-structure preserved by a U(1) action, giving characterizations in the symplectic half-flat and integrable case.

On some examples in Symplectic Topology

Article is devoted to the Examples 2 and 3 of the symplectic solvable Lie groups $R$ with some special cohomological properties, which have been constructed by Benson and Gordon. But they are not succeeded in constructing corresponding compact forms for symplectic structures on these Lie groups. Recently A.Tralle proved that there is no compact form in the Example 3. But his proof is rather complicated and uses some very special topological methods. We propose much more simpler (and purely algebraic) method to prove the main result of the Tralle's paper. Moreover we prove that for Example 2 th…

Symplectic half-flat solvmanifolds

We classify solvable Lie groups admitting left invariant symplectic half-flat structure. When the Lie group has a compact quotient by a lattice, we show that these structures provide solutions of supersymmetric equations of type IIA.

The diversity of symplectic Calabi-Yau six-manifolds

Given an integer b and a finitely presented group G we produce a compact symplectic six-manifold with c_1 = 0, b_2 > b, b_3 > b and fundamental group G. In the simply-connected case we can also arrange for b_3 = 0; in particular these examples are not diffeomorphic to Kähler manifolds with c_1 = 0. The construction begins with a certain orientable four-dimensional hyperbolic orbifold assembled from right-angled 120-cells. The twistor space of the hyperbolic orbifold is a symplectic Calabi-Yau orbifold; a crepant resolution of this last orbifold produces a smooth symplectic manifold with the re…

Strict Quantization of Solvable Symmetric Spaces

This work is a contribution to the area of Strict Quantization (in the sense of Rieffel) in the presence of curvature and non-Abelian group actions. More precisely, we use geometry to obtain explicit oscillatory integral formulae for strongly invariant strict deformation quantizations of a class of solvable symplectic symmetric spaces. Each of these quantizations gives rise to a field of (pre)-C*-algebras whose fibers are function algebras which are closed under the deformed product. The symmetry group of the symmetric space acts on each fiber by C*-algebra automorphisms.

A new six dimensional irreducible symplectic variety

By results of the author there exists a projective (holomorphic) symplectic desingularization of the moduli space of rank-two torsion-free sheaves on a genus-two Jacobian with $c_1=0$ and $c_2=2$. This desingularization has a natural map to the self-product of the Jacobian. We show that the fiber over $(0,0)$ is a 6-dimensional projective irreducible symplectic variety (and hence a 12-dimensional compact Hyperkahler manifold) with second Betti number equal to 8. Thus it is not deformation equivalent to any of the (few) known examples of irreducible symplectic varieties, even up to birational e…

A note on solvable Lie groups without lattices and the Felix-Thomas models of fibrations

In this paper we show that a certain solvable Lie group constructed in a paper by Benson and Gordon has no lattices. This result answers (in the negative way) a question posed by several authors in the context of symplectic geometry. The main theorem is proved with the use of rational homotopy theory.

六维特征可解nil流形上左不变度量的模空间

The moduli space of left-invariant metrics on six-dimensional characteristically solvable nilmanifolds

本文显式确定了三角型特征可解6维nil流形左不变度量的模空间及完整等距群，计算了对称性指数与分布，并发现了首个不存在正对称性指数左不变度量的李群示例，证明了nilsoliton度量能检测此类度量的存在性。

局部共形辛李代数与可解流形的结构

Structure of locally conformally symplectic Lie algebras and solvmanifolds

该研究针对局部共形辛李代数展开结构分析，完成了四维李代数上局部共形辛结构的分类，还在四维连通单连通可解李群的紧商上构造了这类结构。

Mirror Symmetry in Emergent Gravity

Given a six-dimensional symplectic manifold $(M, B)$, a nondegenerate, co-closed four-form $C$ introduces a dual symplectic structure $\widetilde{B} = *C $ independent of $B$ via the Hodge duality $*$. We show that the doubling of symplectic structures due to the Hodge duality results in two independent classes of noncommutative U(1) gauge fields by considering the Seiberg-Witten map for each symplectic structure. As a result, emergent gravity suggests a beautiful picture that the variety of six-dimensional manifolds emergent from noncommutative U(1) gauge fields is doubled. In particular, the…

$\mathcal{F}$-调和3-形式与6维辛李代数上的Type IIA流

$\mathcal{F}$-harmonic 3-forms and the Type IIA flow on 6-dimensional symplectic Lie algebras

本文研究$\mathcal{F}$-调和形式及Type IIA流在6维幂零和可解辛李代数上的长时间行为，证明该流能有效检测拉格朗日环面纤维化和调和近复结构等几何结构。

Canonical almost pseudo-Kähler structures on six-dimensional nilpotent Lie groups

It is known that there are 34 classes of isomorphic connected simply connected six-dimensional nilpotent Lie groups. Of these, only 26 classes suppose left-invariant symplectic structures \cite{Goze-Khakim-Med}. In \cite{CFU2} it is shown that 14 classes of symplectic six-dimensional nilpotent Lie groups suppose compatible complex structures and, therefore, define pseudo-Kähler metrics. In this paper we show that on the remaining 12 classes of six-dimensional nilpotent symplectic Lie groups there are left-invariant almost pseudo-Kähler metrics, and we study their geometrical properties.

Complex, symplectic and Kaehler structures on four dimensional Lie groups

In this work we deal with left invariant complex and symplectic structures on simply connected four dimensional solvable real Lie groups. We search the general form of such structures, when they exist and we make use of this information to determine all left invariant Kaehler structures. Finally, as an appendix we compute explicitly the real cohomology of the corresponding Lie algebras.

Compatibility of balanced and SKT metrics on two-step solvable Lie groups

It has been conjectured by Fino and Vezzoni that a compact complex manifold admitting both a compatible SKT and a compatible balanced metric also admits a compatible Kähler metric. Using the shear construction and classification results for two-step solvable SKT Lie algebras from our previous work, we prove this conjecture for compact two-step solvmanifolds endowed with an invariant complex structure which is either (a) of pure type or (b) of dimension six. In contrast, we provide two counterexamples for a natural generalisation of this conjecture in the homogeneous invariant setting. As part …

Four dimensional symplectic Lie algebras

In this paper we deal with symplectic Lie algebras. All symplectic structures are determined for dimension four and the corresponding Lie algebras are classified up to equivalence. Symplectic four dimensional Lie algebras are described either as solutions of the cotangent extension problem or as symplectic double extension of $\Bbb RR^2$ by $\Bbb RR$. The difference in the choice of a certain model lies on the existence or not of a lagrangian ideal. Moreover all extensions of a two dimensional Lie algebra are determined, and so all solutions (up to equivalence) of the cotangent extension probl…

不满足 hard-Lefschetz 条件的辛可解流形

Symplectic solvmanifolds not satisfying the hard-Lefschetz condition

本文研究 $\R\ltimes_\phi\R^m$ 型完全可解李群，证明作用非半单时其辛可解流形必在次数1或2上违反 hard-Lefschetz 条件，并构造格给出实例。

李代数的中心扩张、动力系统与辛幂零流形

Central extensions of Lie algebras, dynamical systems, and symplectic nilmanifolds

本文研究李代数中心扩张与动力系统的联系，分析幂零李代数无限中心扩张序列的余伴随表示轨道，结合紧幂零流形证明辛幂零流形的覆盖李群作为可解李群可具有任意秩。

Left-invariant almost para-complex structures on six-dimensional nilpotent Lie groups

There are five six-dimensional nilpotent Lie groups G, which do not admit neither symplectic, nor complex structures and, therefore, can be neither almost pseudo-Kahler, nor almost Hermitian. In this work, these Lie groups are being studied. The aim of the paper is to define new left-invariant geometric structures on the Lie groups under consideration that compensate, in some sense, the absence of symplectic and complex structures. New examples of multiparametric families of metrics of signature (3,3) and almost para-complex pseudo-Riemannian half-flat structures on six-dimensional nilmanifold…

关于六维幂零非辛李群上的左不变半凯勒结构

On left-invariant semi-Kähler structures on six-dimensional nilpotent nonsymplectic Lie groups

本文针对三类六维幂零非辛李群，定义并研究了其上的左不变半凯勒结构与近仿半凯勒结构及其几何性质。

On the Existence and Properties of Left Invariant $k$-Symplectic Structures on Lie Groups with Bi-Invariant Pseudo-Riemannian Metric

$k$-symplectic manifolds are a convenient framework to study classical field theories and they are a generalization of polarized symplectic manifolds. This paper focus on the existence and the properties of left invariant $k$-symplectic structures on Lie groups having a bi-invariant pseudo-Riemannian metric. We show that compact semi-simple Lie groups and a large class of Lie groups having a bi-invariant pseudo-Riemannian metric does not carry any left invariant $k$-symplectic structure. This class contains the oscillator Lie groups which are the only solvable non abelian Lie groups having a b…

Special complex manifolds

We introduce the notion of a special complex manifold: a complex manifold (M,J) with a flat torsionfree connection \nabla such that (\nabla J) is symmetric. A special symplectic manifold is then defined as a special complex manifold together with a \nabla-parallel symplectic form ω. This generalises Freed's definition of (affine) special Kähler manifolds. We also define projective versions of all these geometries. Our main result is an extrinsic realisation of all simply connected (affine or projective) special complex, symplectic and Kähler manifolds. We prove that the above three types of sp…

Construction of Lie algebras with special G2-structures

We give a method to obtain new 7-dimensional Lie algebras endowed with closed and coclosed G2-structures starting from 6-dimensional Lie algebras with symplectic half- at SU(3)-structures and half- at SU(3)- structures, respectively. Finally, we describe all the 7-dimensional Lie algebras with a closed G2-structure that are obtained with this method from the 6-dimensional solvable Lie algebras admitting a symplectic half- at SU(3)- structure.

Sur la Conjecture Des Tores Plats

The present work is devoted to compact completely solvable solvmanifolds which admit Kahlerian metrics whose Kahler forms are homogeneous. In particular, we show that such manifolds are diffeomorphic to flat tori. Our proof is based on Dynkin diagrams associated to left invariant closed 2-forms in completely solvable Lie groups. These diagrams are shown to be closely related to Lagrangian foliations,primitive degrees of homogeneous spaces, and geometric structures such as $(\bkR^ n, Aff (\bkR^ n))$-structures on symplectic homogeneous spaces.

Hermitian structures on cotangent bundles of four dimensional solvable Lie groups

We study hermitian structures, with respect to the standard neutral metric on the cotangent bundle $T^*G$ of a 2n-dimensional Lie group $G$, which are left invariant with respect to the Lie group structure on $T^*G$ induced by the coadjoint action. These are in one-to-one correspondence with left invariant generalized complex structures on $G$. Using this correspondence and results of Cavalcanti-Gualtieri and Fernández-Gotay-Gray, it turns out that when $G$ is nilpotent and four or six dimensional, the cotangent bundle $T^*G$ always has a hermitian structure. However, we prove that if $G$ is a…

具有余维二幂零根的可解李代数上的SKT结构

SKT solvable Lie algebras with codimension two nilradical

本文分类了具有余维二幂零根的六维可解李代数上的SKT结构，并给出将SKT幂零李代数扩张为高维可解李代数的一般构造，同时构造了新的紧可解流形例子。

On the symplectic curvature flow for locally homogeneous manifolds

Recently, J. Streets and G. Tian introduced a natural way to evolve an almost-Kähler manifold called the symplectic curvature flow, in which the metric, the symplectic structure and the almost-complex structure are all evolving. We study in this paper different aspects of the flow on locally homogeneous manifolds, including long-time existence, solitons, regularity and convergence. We develop in detail two classes of Lie groups, which are relatively simple from a structural point of view but yet geometrically rich and exotic: solvable Lie groups with a codimension one abelian normal subgroup a…

Two-step solvable SKT shears

We use the shear construction to construct and classify a wide range of two-step solvable Lie groups admitting a left-invariant SKT structure. We reduce this to a specification of SKT shear data on Abelian Lie algebras, and which then is studied more deeply in different cases. We obtain classifications and structure results for $\mathfrak{g}$ almost Abelian, for derived algebra $\mathfrak{g}'$ of codimension 2 and not $J$-invariant, for $\mathfrak{g}'$ totally real, and for $\mathfrak{g}'$ of dimension at most 2. This leads to a large part of the full classification for two-step solvable SKT a…

关于余维数为2的阿贝尔理想的李代数的Streets-Tian猜想

Streets-Tian Conjecture on Lie algebras with codimension $2$ abelian ideals

本文针对李代数含余维数2阿贝尔理想的李复流形，通过显式构造度量与形变路径，验证了Streets-Tian猜想，拓展了该猜想在非凯勒几何中的适用情形。

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 vari…

Differential Gerstenhaber algebras associated to nilpotent algebras

This article provides a complete description of the differential Gerstenhaber algebras of all nilpotent complex structures on any real six-dimensional nilpotent algebra. As an application, we classify all pseudo-Kählerian complex structures on six-dimensional nilpotent algebras such that the differential Gerstenhaber algebra of its complex structure is quasi-isomorphic to that of its symplectic structure. In a weak sense of mirror symmetry, it is a classification of pseudo-Kähler structures on six-dimensional nilpotent algebras whose mirror images are themselves.

Lattices, cohomology and models of six dimensional almost abelian solvmanifolds

We construct lattices on six dimensional not completely solvable almost abelian Lie groups, for which the Mostow condition does not hold. For the corresponding compact quotients, we compute the de Rham cohomology (which does not agree in general with the Lie algebra one) and a minimal model. We show that some of these solvmanifolds admit not invariant symplectic structures and we study formality and Lefschetz properties.

On $k$-para-Kähler Lie algebras a subclass of $k$-symplectic Lie algebras

$k$-Para-Kähler Lie algebras are a generalization of para-Kähler Lie algebras $(k=1)$ and constitute a subclass of $k$-symplectic Lie algebras. In this paper, we show that the characterization of para-Kähler Lie algebras as left symmetric bialgebras can be generalized to $k$-para-Kähler Lie algebras leading to the introduction of two new structures which are different but both generalize the notion of left symmetric algebra. This permits also the introduction of generalized $S$-matrices. We determine then all the $k$-symplectic Lie algebras of dimension $(k+1)$ and all the six dimensional 2-pa…

Almost-complex invariants of families of six-dimensional solvmanifolds

We compute almost-complex invariants $h^{p,0}_{\overline\partial}$, $h^{p,0}_{\text{Dol}}$ and almost-Hermitian invariants $h^{p,0}_{\barδ}$ on families of almost-Kähler and almost-Hermitian $6$-dimensional solvmanifolds. Finally, as a consequence of almost-Kähler identities we provide an obstruction to the existence of a symplectic structure on a given compact almost-complex manifold. Notice that, when $(X,J,g,ω)$ is a compact almost Hermitian manifold of real dimension greater than four, not much is known concerning the numbers $h^{p,q}_{\overline\partial}$.

The geometry of special symplectic representations

We show there is a class of symplectic Lie algebra representations over any field of characteristic not 2 or 3 that have many of the exceptional algebraic and geometric properties of both symmetric three forms in two dimensions and alternating three forms in six dimensions. All nonzero orbits are coisotropic and the covariants satisfy relations generalising classical identities of Eisenstein and Mathews. The main algebraic result is that suitably generic elements of these representation spaces can be uniquely written as the sum of two elements of a naturally defined Lagrangian subvariety. We g…

具有典范丛的非不变平凡化截面的六维复可解流形

Six-dimensional complex solvmanifolds with non-invariant trivializing sections of their canonical bundle

本文旨在对典范丛由非不变全纯截面平凡化的六维复可解流形对应的李代数进行分类，完成了六维可解强幺模李代数的复结构分类，并构造了相关实例。