Romina M. Arroyo, Marina Nicolini
The aim of this article is to study the existence of invariant SKT structures on nilmanifolds. More precisely, we give a negative answer to the question of whether there exist a $k$-step ($k>2$) complex nilmanifold admitting an invariant SKT metric. We also provide a construction which serves as a tool to generate examples of invariant SKT structures on $2$-step nilmanifolds in arbitrary dimensions.
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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.
In this note we observe that on a 2-step nilpotent Lie group equipped with a left-invariant SKT structure the (1,1)-part of the Bismut-Ricci form is seminegative definite. As application we give a simplified proof of the non-existence of invariant SKT static metrics on 2-step nilmanifolds and of the existence of a long time solution to the pluriclosed flow in 2-step nilmanifolds.
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 man…
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) end…
We prove that any invariant strong Kahler structure with torsion (SKT structure) on a flag manifold M=G/K of a semisimple compact Lie group G is Kahler. As an application we describe invariant generalized Kahler structures on M.
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 …
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…
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 …
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。
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…
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 c…
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.
We classify nilmanifolds admitting invariant cocalibrated $G_2$-structures
For a complex manifold $(M,J)$, an SKT (or pluriclosed) metric is a $J$-Hermitian metric $g$ whose fundamental form $ω:=g(J\cdot,\cdot)$ satisfies the condition $\partial\overline{\partial}ω=0$. As such, an SKT metric can be regarded as a natural generalization of a Kähler metric. In this paper, the exceptional Lie group $G_2$ is equipped with a left-invariant integrable almost complex structure $\mathcal{J}$ via the Samelson construction and a 7-parameter family of $\mathcal{J}$-Hermitian metrics is constructed. From this 7-parameter family, the members which are SKT are calculated. The resul…
Hermitian structures on six dimensional nilmanifolds
设（J，g）是紧致流形M上的Hermite结构，具有不变的复结构J和相容度量g，g不要求是不变的.给出了具有挠、平衡或局部共形Kähler结构（J，g）的强Kähler的6维幂零流形M的分类.
A strong KT (SKT) manifold consists of a Hermitian structure whose torsion three-form is closed. We classify the invariant SKT structures on four-dimensional solvable Lie groups. The classification includes solutions on groups that do not admit compact four-dimensional quotients. It also shows that there are solvable groups in dimension four that admit invariant complex structures but have no invariant SKT structure.
Closed Geodesics on Compact Nilmanifolds with Chevalley Rational Structure
我们继续研究了由单连通二步幂零李群和格构成的具有左不变度量的幂零流形上的闭测地线的分布。考虑实有限维向量空间上紧致半单李代数的不可约表示所构造的李群及其2步幂零李代数.本文在半单李代数上确定了由Chevalley有理结构产生的格的零流形具有闭测地线密度性质的充分条件。
CYT and SKT Metrics on Compact Semi-Simple Lie Groups
本文研究了赋予Samelson复结构的紧半单李群上左不变SKT和CYT度量的存在性，证明了在特定不变性条件下同时满足CYT和SKT的度量必为Bismut平坦。
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 $G_{2(2)}^*$.
Let $(X,J)$ be a nilmanifold with a left-invariant nilpotent complex structure. We study the existence of $p$-Kähler structures (which include Kähler and balanced metrics) on $X$. More precisely, we determine an optimal $p$ such that there are no $p$-Kähler structures on $X$. Finally, we show that, contrarily to the Kähler case, on compact complex manifolds there is no relation between the existence of balanced metrics and the degeneracy step of the Frölicher spectral sequence. More precisely, on balanced manifolds the degeneracy step can be arbitrarily large.
We use Bott-Chern cohomology to measure the non-Kählerianity of 6-dimensional nilmanifolds endowed with the invariant complex structures in M. Ceballos, A. Otal, L. Ugarte, and R. Villacampa's classification, [Invariant Complex Structures on 6-Nilmanifolds: Classification, Frölicher Spectral Sequence and Special Hermitian Metrics, J. Geom. Anal. (2014)]. We investigate the existence of pluriclosed metric in connection with such a classification.
Almost Hermitian Geometry on Six Dimensional Nilmanifolds
用SO（4）xU（1）在复射影三维空间上的作用描述了6维李群上不变的几乎厄米特结构的基本2-形式.这导致了组合描述类几乎厄米结构的岩泽和其他nilmanifold。
In this paper we consider closed $\text{SL}(3,\mathbb{C})$-structures which are either mean convex or tamed by a symplectic form. These notions were introduced by Donaldson in relation to $\text{G}_2$-manifolds with boundary. In particular, we classify nilmanifolds which carry an invariant mean convex closed $\text{SL}(3,\mathbb{C})$-structure and those which admit an invariant mean convex half-flat $\text{SU}(3)$-structure. We also prove that, if a solvmanifold admits an invariant tamed closed $\text{SL}(3,\mathbb{C})$-structure, then it also has an invariant symplectic half-flat $\text{SU}(3…
Dolbeault Cohomology of compact Nilmanifolds
设$M= G/Γ$是一个具有不变复结构的紧致零流形.证明了在$M$上具有不变复结构的模空间${\cal C} ({\frak g})$的任意连通分支的开集上，$M$的Dolbeault上同调同构于与$G$的李代数的复化$\cg^\C$相关的微分双阶化代数的Dolbeault上同调.为了得到这一结果，我们首先证明了具有有理不变式复结构的紧致流形的上述同构。这是使用一个递减系列相关联的复结构和博雷尔谱序列的相应的一组全纯纤维化。然后我们将Kodaira-Spencer理论应用于复结构的变形。
We study generalized complex cohomologies of generalized complex structures constructed from certain symplectic fibre bundles over complex manifolds. We apply our results in the case of left-invariant generalized complex structures on nilmanifolds and to their space of small deformations.
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 a…
The invariant balanced Hermitian geometry of nilmanifolds of dimension 6 is described. We prove that the holonomy group of the associated Bismut connection reduces to a proper subgroup of SU(3) if and only if the complex structure is abelian. As an application we show that if J is abelian then any invariant balanced J-Hermitian structure provides solutions of the Strominger system.
We introduce a double complex that can be associated to certain Lie algebras, and show that its cohomology determines an obstruction to the existence of a half-flat SU(3)-structure. We obtain a classification of the 6-dimensional nilmanifolds carrying an invariant half-flat structure.
$p$-Kähler structures on nilmanifolds and holomorphically parallelizable manifolds
本文证明Alessandrini-Bassanelli猜想，分类了低p值下容许p-Kähler结构的全纯平行可解流形，并给出具有约化万有覆盖的紧全纯平行可流形中(n-2)-Kähler流形的新例子。
Moduli spaces of (co)closed $\mathrm{G}_2$-structures on nilmanifolds
本文计算了7维幂零流形上闭与余闭 $\mathrm{G}_2$ 结构模空间的维数，发现其与第三贝蒂数无关，并证明余闭结构的自同构群未必交换。
We introduce spin-harmonic structures, a class of geometric structures on Riemannian manifolds of low dimension which are defined by a harmonic unitary spinor. Such structures are related to SU(2) (dim=4,5), SU(3) (dim=6) and G_2 (dim=7) structures; in dimension 8, a spin-harmonic structure is equivalent to a balanced Spin(7) structure. As an application, we obtain examples of compact 8-manifolds endowed with non-integrable Spin(7) structures of balanced type.
We study generalized complex structures and $T$-duality (in the sense of Bouwknegt, Evslin, Hannabuss and Mathai) on Lie algebras and construct the corresponding Cavalcanti and Gualtieri map. Such a construction is called "Infinitesimal $T$-duality". As an application we deal with the problem of finding symplectic structures on 2-step nilpotent Lie algebras. We also give a criteria for the intregability of the infinitesimal $T$-duality of Lie algebras to topological $T$-duality of the associated nilmanifolds.
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 parti…
In this note we prove that any left-invariant almost Hermitian structure on a 2-step nilmanifold is Ricci-flat with respect to the Chern connection and that it is Ricci-flat with respect to another canonical connection if and only if it is cosymplectic.
On astheno-Kähler nilmanifolds with balanced metrics
研究复幂零流形上弱Kähler、强Gauduchon和平衡度量的存在性，证明在复维4时弱Kähler度量存在则幂零流形最多2步且第一Betti数≥6，并构造了任意复维n≥4上同时具有弱Kähler和平衡度量的例子。
A cohomology theory associated to a holomorphic Poisson structure is the hypercohomology of a bi-complex where one of the two operators is the classical $\overline\partial$-operator, while the other operator is the adjoint action of the Poisson bivector with respect to the Schouten-Nijenhuis bracket. The first page of the associated spectral sequence is the Dolbeault cohomology with coefficients in the sheaf of germs of holomorphic polyvector fields. In this note, the authors investigate the conditions for which this spectral sequence degenerates on the first page when the underlying complex m…
We introduce obstructions to the existence of a calibrated G_2-structure on a Lie algebra g of dimension seven, not necessarily nilpotent. In particular, we prove that if there is a Lie algebra epimorphism from g to a six-dimensional Lie algebra h with kernel contained in the center of g, then h has a symplectic form. As a consequence, we obtain a classification of the nilpotent Lie algebras that admit a calibrated G_2-structure.
Almost Hermitian geometry and generalized Ricci flow
本文研究全斜对称Nijenhuis张量且Bismut挠率闭合的殆埃尔米特流形（almost-SKT），引入超定椭圆性条件保证受约束Bismut–Ricci流短时存在，并在六维紧流形上构造非可积例子，揭示其与广义Ricci流的联系。
Let (N,J) be a real 2n-dimensional nilpotent Lie group endowed with an invariant complex structure. A left-invariant Riemannian metric on N compatible with J is said to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics on (N,J) with the same scalar curvature. In this paper, we determine all complex structures that admit a minimal compatible metric on 6-dimensional nilpotent Lie groups.
On the nu-invariant of two-step nilmanifolds with closed G2-structure
研究了具有闭G2结构的二步nilmanifolds上的nu不变量，通过计算调和旋量和η不变量，确定了其存在性和奇偶性，并推导出在不变调和旋量上nu的消失。
Purely coclosed $\mathrm{G}_2$-structures on nilmanifolds -- II
本文完成七维 nilpotent 利格群上左不变纯 coclosed $\mathrm{G}_2$-结构的分类，针对不可分解的 5-和 6-步 nilpotent 利格群进行研究。
We study the interplay between the following types of special non-Kähler Hermitian metrics on compact complex manifolds: it locally conformally Kähler, $k$-Gauduchon, balanced and locally conformally balanced and prove that a locally conformally Kähler compact nilmanifold carrying a balanced or a $k$-Gauduchon metric is necessarily a torus. Combined with a result of Fino and Vezzoni from 2016, this leads to the fact that a compact complex 2-step nilmanifold endowed with whichever two of the following types of metrics: balanced, pluriclosed and locally conformally Kähler is a torus. Moreover, w…
The Bott-Chern cohomology of 6-dimensional nilmanifolds endowed with invariant complex structure is studied with special attention to the cases when balanced or strongly Gauduchon Hermitian metrics exist. We consider complex invariants introduced by Angella and Tomassini and by Schweitzer, which are related to the $\partial \bar\partial$-lemma condition and defined in terms of the Bott-Chern cohomology, and show that the vanishing of some of these invariants is not a closed property under holomorphic deformations. In the balanced case, we determine the spaces that parametrize deformations in t…
We answer in the affirmative the question posed by Conti and Rossi on the existence of nilpotent Lie algebras of dimension 7 with an Einstein pseudo-metric of nonzero scalar curvature. Indeed, we construct a left-invariant pseudo-Riemannian metric $g$ of signature $(3, 4)$ on a nilpotent Lie group of dimension 7, such that $g$ is Einstein and not Ricci-flat. We show that the pseudo-metric $g$ cannot be induced by any left-invariant closed $G_2^*$-structure on the Lie group. Moreover, some results on closed and harmonic $G_2^*$-structures on an arbitrary 7-manifold $M$ are given. In particular,…
In this paper we give the list of all 7-dimensional nilpotent real Lie algebras that admit a contact structure. Based on this list, we describe all 7-dimensional nilmanifolds that admit an invariant contact structure.
Invariant forms compute the Dolbeault cohomology of complex nilmanifolds
该研究证明赋予左不变复结构的紧幂零流形的左不变形式到Dolbeault复形的包含映射在各双次数上诱导上同调同构，还解决了Hasegawa、Angella等人的相关猜想，明确了三类不变量的计算方式与独立性。
Vaisman Solvmanifolds as Finite Quotients of Kodaira-Thurston Nilmanifolds
本文证明了所有 Vaisman 线性流形都是 Kodaira-Thurston 流形的有限商，并探讨了相关流形的结构与性质。
Generalized complex structures on nilmanifolds
本文证明所有6维幂零流形均容许广义复结构，并按其容许的左不变广义复结构类型完成分类，同时发现Iwasawa流形的复模空间分支在广义复模空间中相交，并构造了一个不容许左不变广义复结构的8维幂零流形。
We discuss the known evidence for the conjecture that the Dolbeault cohomology of nilmanifolds with left-invariant complex structure can be computed as Lie-algebra cohomology and also mention some applications.
We give a construction of integrable complex structures on the total space of a smooth principal bundle over a complex manifold, with an even dimensional compact Lie group as structure group, under certain conditions. This generalizes the constructions of complex structure on compact Lie groups by Samelson and Wang, and on principal torus bundles by Calabi-Eckmann and others. It also yields large classes of new examples of non-Kähler compact complex manifolds. Moreover, under suitable restrictions on the base manifold, the structure group, and characteristic classes, the total space of the pri…
It is proved that on nilmanifolds with abelian complex structure, there exists a canonically constructed non-trivial holomorphic Poisson structure. We identify the necessary and sufficient condition for its associated cohomology to be isomorphic to the cohomology associated to trivial (zero) holomorphic Poisson structure. We also identify a sufficient condition for this isomorphism to be at the level of Gerstenhaber algebras.
Deformation of 2-step Nilmanifolds with Abelian Complex Structures
我们发展了零流形上阿贝尔不变复结构的形变理论，并证明了在这种情况下，Kuranishi过程保持不变性质。一个纯粹的代数条件的特点再次导致阿贝尔结构的变形，我们证明，这样的变形是通畅的。各种例子说明了由此产生的理论，以及在3个复杂的维度中可能的行为。
Geometry of nilmanifolds with left-invariant complex structure and deformations in the large
阐明了具有左不变复结构的零流形与迭代主全纯环面丛之间的关系，并给出了大变形也是这种类型的判别准则。作为一个应用程序，我们得到一个相当完整的图片在复杂的三维。
Symplectic structures on low dimensional 2-step nilmanifolds
本文研究低维2步nilmanifolds上的辛结构，证明了存在II型闭2-形式是获得辛结构的必要条件，且在低维多数情况下该条件也是充分的。
G_2-structures with torsion from half-integrable nilmanifolds
本文研究了乘积M^7 = N^6 × S^1上有挠的G_2-结构方程与N^6上诱导的SU（3）-结构的关系。当G_2-结构的Lee-形式不为零且N^6是一个具有半可积SU（3）-结构的六维零流形时，所有的解都被找到.讨论了这些解的挠率的特殊性质。
We classify invariant complex structures on 6-dimensional nilmanifolds up to equivalence. As an application, the behaviour of the associated Frölicher sequence is studied as well as its relation to the existence of strongly Gauduchon metrics. We also show that the strongly Gauduchon property and the balanced property are not closed under holomorphic deformation.
Lie-algebra Dolbeault cohomology and small deformations of nilmanifolds
我们考虑具有左不变复结构的零流形，并证明了如果零流形的Dolbeault上同调可以用左不变形式计算，那么这种结构的小变形也是左不变的.根据Console和Fino的结果，通常情况下是这样的。我们的主要工具是具有复结构的李代数的Dolbeault-上同调的一个模拟。
Canonical bundles of complex nilmanifolds, with applications to hypercomplex geometry
一个幂零群$G$与一个余紧离散子群的商称为一个幂零流形。复零流形是具有$G$不变复结构的零流形.我们证明了复零流形有平凡的典范丛。这是用来研究超复流形（具有满足四元数关系的$G$ -不变复结构的三元组的流形）。我们证明了超复零流形允许HKT度量当且仅当其超复结构是阿贝尔的。此外，任何$G$ -不变的HKT-度量的零流形是平衡的所有相关的复结构。
We prove that if a compact nilmanifold $Γ\backslash G$ is endowed with a Vaisman structure, then $G$ is isomorphic to the Cartesian product of the Heisenberg group with $\mathbb{R}$.
We study bi-Lagrangian structures (a symplectic form with a pair of complementary Lagrangian foliations, also known as para-Kähler or Künneth structures) on nilmanifolds of dimension less than or equal to 6. In particular, building on previous work of several authors, we determine which 6-dimensional nilpotent Lie algebras admit a bi-Lagrangian structure. In dimension 6, there are (up to isomorphism) 26 nilpotent Lie algebras which admit a symplectic form, 16 of which admit a bi-Lagrangian structure and 10 of which do not. We also calculate the curvature of the canonical connection of these bi…
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