Hermitian structures on six dimensional nilmanifolds

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。

Complex nilmanifolds and Kähler-like connections

In this note, we analyze the question of when will a complex nilmanifold have Kähler-like Strominger (also known as Bismut), Chern, or Riemannian connection, in the sense that the curvature of the connection obeys all the symmetries of that of a Kähler metric. We give a classification in the first two cases and a partial description in the third case. It would be interesting to understand these questions for all Lie-Hermitian manifolds, namely, Lie groups equipped with a left invariant complex structure and a compatible left invariant metric.

Complex nilmanifolds with constant holomorphic sectional curvature

A well known conjecture in complex geometry states that a compact Hermitian manifold with constant holomorphic sectional curvature must be Kähler if the constant is non-zero and must be Chern flat if the constant is zero. The conjecture is confirmed in complex dimension $2$, by the work of Balas-Gauduchon in 1985 (when the constant is zero or negative) and by Apostolov-Davidov-Muskarov in 1996 (when the constant is positive). For higher dimensions, the conjecture is still largely unknown. In this article, we restrict ourselves to the class of complex nilmanifolds and confirm the conjecture in …

Almost Hermitian Structures with Parallel Torsion

The characteristic connection of an almost hermitian structure is a hermitian connection with totally skew-symmetric torsion. The case of parallel torsion in dimension six is of particular interest. In this work, we give a full classification of the algebraic types of the torsion form, and, based on this, undertake a systematic investigation into the possible geometries. Numerous naturally reductive spaces are constructed and classified as well as examples on nilmanifolds given.

A family of complex nilmanifolds with infinitely many real homotopy types

We find a one-parameter family of non-isomorphic nilpotent Lie algebras $\mathfrak{g}_a$, with $a \in [0,\infty)$, of real dimension eight with (strongly non-nilpotent) complex structures. By restricting $a$ to take rational values, we arrive at the existence of infinitely many real homotopy types of $8$-dimensional nilmanifolds admitting a complex structure. Moreover, balanced Hermitian metrics and generalized Gauduchon metrics on such nilmanifolds are constructed.

Compatibility between non-Kähler structures on complex (nil)manifolds

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…

关于具有平衡度量的弱Kähler幂零流形

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。

The Anomaly flow on nilmanifolds

We study the Anomaly flow on $2$-step nilmanifolds with respect to any Hermitian connection in the Gauduchon line. In the case of flat holomorphic bundle, the general solution to the Anomaly flow is given for any initial invariant Hermitian metric. The solutions depend on two constants $K_1$ and $K_2$, and we study the qualitative behaviour of the Anomaly flow in terms of their signs, as well as the convergence in Gromov-Hausdorff topology. The sign of $K_1$ is related to the conformal invariant introduced by Fu, Wang and Wu. In the non-flat case, we find the general evolution equations of the…

Nilmanifolds with a calibrated G_2-structure

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.

Dolbeault cohomology of complex nilmanifolds foliated in toroidal groups

It is conjectured that the Dolbeault cohomology of a complex nilmanifold $X$ is computed by left-invariant forms. We prove this under the assumption that $X$ is suitably foliated in toroidal groups and deduce that the conjecture holds in real dimension up to six. Our approach generalises previous methods, where the existence of a holomorphic fibration was a crucial ingredient.

Balanced Hermitian metrics from SU(2)-structures

We study the intrinsic geometrical structure of hypersurfaces in 6-manifolds carrying a balanced Hermitian SU(3)-structure, which we call {\em balanced} SU(2)-{\em structures}. We provide conditions which imply that such a 5-manifold can be isometrically embedded as a hypersurface in a manifold with a balanced SU(3)-structure. We show that any 5-dimensional compact nilmanifold has an invariant balanced SU(2)-structure as well as new examples of balanced Hermitian SU(3)-metrics constructed from balanced SU(2)-structures. Moreover, for $n=3,4$, we present examples of compact manifolds, endowed w…

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…

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.

G_2-structures with torsion from half-integrable nilmanifolds

The equations for a G_2-structure with torsion on a product M^7 = N^6 x S^1 are studied in relation to the induced SU(3)-structure on N^6. All solutions are found in the case when the Lee-form of the G_2-structure is non-zero and N^6 is a six-dimensional nilmanifold with half-integrable SU(3)-structure. Special properties of the torsion of these solutions are discussed.

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.

具有常数混合曲率的厄米流形

On Hermitian manifolds with constant mixed curvature

本文验证了Kai Tang关于非零常数混合曲率紧致厄米流形必为凯勒流形的猜想，针对复幂零流形、可解流形等特殊类型及平衡三维流形，提供了部分证据。

Nilmanifolds and their associated non local fields

For six dimensional nilmanifolds we build a module $\mathcal{H}$ of an affine Kac Moody vertex algebras. Then, we associate some logarithmic fields for the module $\mathcal{H}$ and we study their singularities. We also presented a physics motivation behind this construction. We study a particular case, we show that when the nilmanifold $N$ is a $k$ degree $S^1$--fibration over the two torus and a choice of $l \in \mathbb{Z} \simeq H^3(N, \mathbb{Z})$ the fields associated to the space $\mathcal{H}$ have tri-logarithm singularities whenever $kl \neq 0$.

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 nilmanifo…

Families of strong KT structures in six dimensions

This paper classifies Hermitian structures on 6-dimensional nilmanifolds M=G/L for which the fundamental 2-form is d d-bar closed, a condition that is shown to depend only on the underlying complex structure J of M. The space of such J is described when G is the complex Heisenberg group, and explicit solutions are obtained from a limacon-shaped curve in the complex plane. Related theory provides examples of various types of Ricci-flat structures.

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

In this paper we consider left-invariant pseudo-Kähler structures on six-dimensional nilpotent Lie algebras. The explicit expressions of the canonical complex structures are calculated, and the curvature properties of the associated pseudo-Kähler metrics are investigated. It is proved that the associated pseudo-Kähler metric is Ricci-flat, that the curvature tensor has zero pseudo-Riemannian norm, and that the curvature tensor has some non-zero components that depend only on two or, at most, three parameters. The pseudo-Kähler structures obtained give basic models of pseudo-Kähler six-dimensio…

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…

SageMath experiments in Differential and Complex Geometry

This note summarizes the talk by the author at the workshop "Geometry and Computer Science" held in Pescara in February 2017. We present how SageMath can help in research in Complex and Differential Geometry, with two simple applications, which are not intended to be original. We consider two "classification problems" on quotients of Lie groups, namely, "computing cohomological invariants" [D. Angella, M. G. Franzini, F. A. Rossi, Degree of non-Kählerianity for 6-dimensional nilmanifolds, Manuscripta Math. 148 (2015), no. 1-2, 177--211], [A. Latorre, L. Ugarte, R. Villacampa, On the Bott-Chern…

具有特殊 Hermite 度量的紧复流形的 Frölicher 谱序列

Frölicher spectral sequence of compact complex manifolds with special Hermitian metrics

研究紧复流形上 balanced、SKT 与广义 Gauduchon 度量和 Frölicher 谱序列退化行为的关系，构造任意页不退化的无穷族并给出否定 Popovici 猜想的 SKT 反例。

关于具有复换位子和常数全纯截面曲率的解流形

On solvmanifolds with complex commutator and constant holomorphic sectional curvature

本文针对非凯勒几何中“具常数全纯截面曲率的紧埃尔米特流形必为凯勒或陈平坦”的猜想，在具复换位子的解流形类上完成证明，拓展了此前幂零流形相关的研究结果。

六维殆幂零可解流形上的埃尔米特结构

Hermitian structures on six-dimensional almost nilpotent solvmanifolds

本研究完成了六维强幺模殆幂零李代数的复结构分类，确定了对应可解流形的不变复结构及多种特殊埃尔米特度量的存在性，证实了相关猜想并证明了辛驯复结构的部分不存在性结果。

两步幂零李群上的复结构

Complex structures on two-step nilpotent Lie groups

本文刻画了容许左不变复结构的 $2$ 步幂零李代数，研究了其 Hermitian 几何，给出了度量为 pluriclosed 的充要条件，并将结果推广至 hypercomplex 情形。

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) end…

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 $h^{p,q}_{\overline\partial}$.

Stability of Abelian Complex Structures

Let $M = Γ\backslash G$ be a nilmanifold endowed with an invariant complex structure. We prove that Kuranishi deformations of abelian complex structures are all invariant complex structures, generalizing a result of C. Maclaughlin, H. Pedersen, Y.S. Poon and S. Salamon for 2-step nilmanifolds. We characterize small deformations that remain abelian. As an application, we observe that at real dimension six, the deformation process of abelian complex structures is stable within the class of nilpotent complex structures. We give an example to show that this property does not hold in higher dimensi…

论低维 Solvmanifold（可解流形）

On Low-Dimensional Solvmanifolds

本文研究六维及以下可解李群中格的存在性，并系统分类相应可解流形的形式性、辛、Kähler 及（Hard）Lefschetz 性质。

Quaternion geometries on the twistor space of the six-sphere

We explicitly describe all SO(7)-invariant almost quaternion-Hermitian structures on the twistor space of the six sphere and determine the types of their intrinsic torsion.

具有二维交换子群的李群上的正交结构

Hermitian and Kähler Structures of Two Types on Lie Groups with Two-Dimensional Derived Algebras

本文研究了具有二维交换子群的李群上的左不变正交结构，分类了两种类型，并计算了凯勒结构的比姆特连接，同时给出了相关例子。

Curvature, integrability, and the six sphere

This note is about the interplay between the almost-hermitian and Riemannian geometries of a manifold. These geometries can be seen to interact through curvature. The main result is an obstruction equation to the integrability of almost-complex structures orthogonal with respect to Riemannian metrics with constrained sectional curvature. Several geometric consequences ensue, such as a formula for the norm of the Levi-Civita covariant derivative of a hypothetical orthogonal complex structure. Our results lead to a partial recovery of the well-known fact that the round $6$-sphere $S^6$ is not he…

Hermitian structures on a class of almost nilpotent solvmanifolds

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…

关于复结构与全纯向量丛对的形变分裂

On splittings of deformations of pairs of complex structures and holomorphic vector bundles

本文研究复流形与全纯向量丛对的Kuranishi空间分裂问题，证明Kähler情形和复可平行化nilmanifold平凡丛情形下可分解为直积，并给出非分裂的nilmanifold反例。

The moduli space of left-invariant metrics of a class of six-dimensional nilpotent Lie groups

In this paper we determine the moduli space, up to isometric automorphism, of left-invariant metrics on a $6$-dimensional Lie group $H$, such that its Lie algebra $\mathfrak{h}$ admits a complex structure and has first Betti number equal to four. We also investigate which of these metrics are Hermitian and classify the corresponding complex structures.

近厄米特流形与凯勒流形上的四分之一对称联络

Quarter-symmetric connection on an almost Hermitian manifold and on a Kähler manifold

本文研究几乎厄米特流形上四分之一对称联络的性质，证明保持广义黎曼度量的情形等价于凯勒流形，并构造了与联络生成元无关的曲率张量，揭示其与Weyl射影曲率张量及全纯射影曲率张量的关系。

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…

Associated Nijenhuis Tensors on Manifolds with Almost Hypercomplex Structures and Metrics of Hermitian-Norden Type

An associated Nijenhuis tensor of endomorphisms in the tangent bundle is introduced. Special attention is paid to such tensors for an almost hypercomplex structure and the metric of Hermitian-Norden type. There are studied relations between the six associated Nijenhuis tensors as well as their vanishing. It is given a geometric interpretation of the vanishing of these tensors as a necessary and sufficient condition for the existence of a unique connection with totally skew-symmetric torsion tensor. Similar idea is used in the paper of T. Friedrich and S. Ivanov in Asian J. Math. (2002) for som…

Balanced Hermitian structures on almost abelian Lie algebras

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. Bedull…

扭曲笛卡尔积上的平衡埃尔米特结构

Balanced Hermitian structures on twisted cartesian products

研究两个埃尔米特李代数根据表示ρ1和ρ2的扭曲笛卡尔积上的埃尔米特结构，给出其平衡和局部共形平衡的条件，并分类六维平衡埃尔米特扭曲笛卡尔积李代数。

殆埃尔米特几何与广义Ricci流

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。

关于余齐一六维流形上平衡度量的存在性

On the existence of balanced metrics on six-manifolds of cohomogeneity one

本文证明了单连通六维流形上余齐一作用下不变的平衡非凯勒SU(3)结构不存在，为Hull-Strominger系统提供了理论限制。

Griffiths positivity for Bismut curvature and its behaviour along Hermitian Curvature Flows

In this note we study a positivity notion for the curvature of the Bismut connection; more precisely, we study the notion of \emph{Bismut-Griffiths-positivity} for complex Hermitian non-Kähler manifolds. Since the Kähler-Ricci flow preserves and regularizes the usual Griffiths positivity we investigate the behaviour of the Bismut-Griffiths-positivity under the action of the Hermitian curvature flows. In particular we study two concrete classes of examples, namely, linear Hopf manifolds and six-dimensional Calabi-Yau solvmanifolds with holomorphically-trivial canonical bundle. From these exampl…

Complex structures of splitting type

We study the six-dimensional solvmanifolds that admit complex structures of splitting type classifying the underlying solvable Lie algebras. In particular, many complex structures of this type exist on the Nakamura manifold $X$, and they allow us to construct a countable family of compact complex non-$\partial\overline\partial$ manifolds $X_k$, $k\in\mathbb{Z}$, that admit a small holomorphic deformation $\{(X_{k})_{t}\}_{t\inΔ_k}$ satisfying the $\partial\overline\partial$-Lemma for any $t\inΔ_k$ except for the central fibre. Moreover, a study of the existence of special Hermitian metrics is …

Generalized Kähler almost abelian Lie groups

We study left-invariant generalized Kähler structures on almost abelian Lie groups, i.e., on solvable Lie groups with a codimension-one abelian normal subgroup. In particular, we classify six-dimensional almost abelian Lie groups which admit a left-invariant complex structure and establish which of those have a left-invariant Hermitian structure whose fundamental 2-form is $\partial \bar \partial$-closed. We obtain a classification of six-dimensional generalized Kähler almost abelian Lie groups and determine the 6-dimensional compact almost abelian solvmanifolds admitting an invariant generali…

具有全纯平凡典范丛的六维齐性空间

Six dimensional homogeneous spaces with holomorphically trivial canonical bundle

本研究分类了6维幺模李代数，得到具有全纯平凡典范丛的紧齐性空间，还证明了紧非凯勒平衡齐性空间中Hull-Strominger系统不变解的唯一性，并在Nakamura流形上构造了杂弦运动方程的解。

Homogeneous nearly Kähler manifolds

We classify six-dimensional homogeneous nearly Kähler manifolds and give a positive answer to Gray and Wolf's conjecture: every homogeneous nearly Kähler manifold is a Riemannian 3-symmetric space equipped with its canonical almost Hermitian structure. The only four examples in dimension 6 are $S^3 \times S^3$, the complex projective space $\CM P^3$, the flag manifold $\mathbb F^3$ and the sphere $S^6$. We develop, about each of these spaces, a distinct aspect of nearly Kähler geometry and make in the same time a sharp description of its specific homogeneous structure.

关于 S^6 上广义近复结构的可积性

On the integrability of generalized almost complex structures on $\mathbb{S}^6$

研究六维球面 S^6 上广义近复结构的可积性，分别考虑由仿射联络确定的括号可积性和强广义近复结构的 Courant 可积性，并证明除 J_g 外无平凡球面组合是 Levi-Civita 联络可积的，且通过粘合构造给出了 Courant 可积的障碍结果。