Dmitri V. Alekseevsky, Liana David
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.
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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.
Existence of strong Kähler with torsion metrics, shortly SKT metrics, on complex manifolds has been shown to be unstable under small deformations. We find necessary conditions under which the property of being SKT is stable for a smooth curve of Hermitian metrics $\{ω_t\}_t$ which equals a fixed SKT metric $ω$ for $t=0$, along a differentiable family of complex manifolds $\{M_t\}_t$.
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。
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…
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 study the existence of three classes of Hermitian metrics on certain types of compact complex manifolds. More precisely, we consider balanced, SKT and astheno-Kähler metrics. We prove that the twistor spaces of compact hyperkähler and negative quaternionic-Kähler manifolds do not admit astheno-Kähler metrics. Then we provide examples of astheno-Kähler structures on toric bundles over Kähler manifolds. In particular, we find examples of compact complex non-Kähler manifolds which admit a balanced and an astheno-Kähler metrics, thus answering to a question in [52] (see also [24]). One of these…
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.
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…
A Variational Approach to SKT and Balanced Metrics
本文针对SKT与平衡度量定义新泛函，证明其临界点恰为Kähler度量，为Kähler性障碍研究提供新途径。
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 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…
SKT Hyperbolic and Gauduchon Hyperbolic Compact Complex Manifolds
本文为紧复流形引入SKT双曲性与Gauduchon双曲性，证明前者蕴含Kobayashi/Brody双曲性，后者蕴含除子双曲性，并给出SKT双曲流形万有覆盖上L^2调和空间的消失定理。
Bott-Chern formality and Massey products on strong Kähler with torsion and Kähler solvmanifolds
研究几何 Bott-Chern 形式性与 SKT 度量及 Kähler 可解流形的关系，证明 6 维幂零流形存在 SKT 度量等价于几何 Bott-Chern 形式性，并构造非零 ABC-Massey 积。
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…
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…
The equality between the balanced and the Gauduchon cones is discussed in several situations. In particular, it is shown that equality does not hold on many twistor spaces, and it holds on Moishezon manifolds. Moreover, it is proved that a SKT manifold of dimension three on which the balanced cone equals the Gauduchon cone is in fact Kahler.
A Comparative Study between Kähler and Non-Kähler Hyperbolicity
本文建立了SKT与平衡双曲性之间的联系，并揭示了它们与Gromov意义下的Kähler双曲性之间的关系。
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。
Balanced metrics and Gauduchon cone of locally conformally Kahler manifolds
本文研究了局部共形Kahler流形的度量性质，提出并部分证明了容许平衡、SKT与LCK三种Hermitian形式中两种的紧复流形必容许Kahler度量的猜想，同时证明了关于Bott--Chern同调的猜想并验证了所有已知LCK流形。
Curvature positivity for Kähler and quasi-Kähler flag manifolds
本文研究Kähler与拟Kähler旗流形及复射影空间上Chern联络曲率的Griffiths与dual-Nakano正性，证明了相关度量的存在性，给出了半正曲率情形的完整分类，并对非可积近复结构提出猜想。
In this article the framework created by Cartan to produce local differential invariants for submanifolds of homogeneous spaces is applied to classify all totally geodesic Lagrangian submanifolds and all homogeneous Lagrangian submanifolds of the nearly Kähler manifold of full flags in $\mathbb{C}^3$.
In this paper, we describe invariant twisted Kähler-Einstein (tKE) metrics on flag varieties. We also explore some applications of the ideas involved in the proof of our main result to the existence of invariant twisted constant scalar curvature Kähler metrics. Also, we provide a precise description for the greatest Ricci lower bound of an arbitrary Kähler class on a flag variety. By means of this description, we establish some inequalities related to optimal volume upper bounds for Kähler metrics just using tools from Lie theory. Further, we describe the set of tKE metrics for several example…
Geometry of Invariant Almost Semi Kähler Submanifolds of Flag Manifolds
该文研究旗流形的不变近半Kähler子流形几何，证明了作为其近Hermite子流形的齐性空间必为极小子流形，若这类子流形也是旗流形则是全测地的。
A product of Kähler manifolds also carries a Kähler metric. In this short note we would like to study the product of generalized $p-$Kähler manifolds, compact or not. The results we get extend the known results (balanced, SKT, sG manifolds), and are optimal in the compact case. Hence we can give new non-trivial examples of generalized $p-$Kähler manifolds.
Invariant scalar-flat Kähler metrics on line bundles over generalized flag varieties
针对广义旗流形上线丛的凯勒度量问题，通过证明线丛全空间的$G$-不变凯勒度量由卡拉比拟设生成，得出每个凯勒类中存在唯一$G$-不变量平坦凯勒度量的结论。
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…
In this paper we provide an explicit description of normal almost contact structures obtained from Cartan-Ehresmann connections (gauge fields) on principal $S^{1}$-bundles over complex flag manifolds. The main feature of our approach is to employ elements of representation theory of complex simple Lie algebras in order to describe and classify these structures. We use these normal almost contact structures to explicitly describe a huge class of compact Hermitian non-Kähler manifolds obtained from products of principal $S^{1}$-bundles over complex flag manifolds. Moreover, we obtain from our de…
We introduce integrable complex structures on twistor spaces fibered over complex manifolds. We then show, in particular, that the twistor spaces associated with generalized Kahler, SKT and strong HKT manifolds all naturally admit complex structures. Moreover, in the strong HKT case we construct a metric and three compatible complex structures on the twistor space that have equal torsions.
Given a simply connected compact generalized flag manifold M together with its invariant Kähler Einstein metric g, we investigate the functional given by the first eigenvalue of the Hodge Laplacian on smooth functions restricted to the space of invariant Kähler metrics. We give sufficient and necessary conditions so that the metric g is a critical point for this functional. Moreover we prove that when M is a full flag manifold, the metric g is critical if and only if M= SU(3)/T^2 and in this case g is a maximum.
In this paper we construct a family of complex structures on a complex flag manifold that converge to the real polarization coming from the Gelfand-Cetlin integrable system, in the sense that holomorphic sections of a prequantum line bundle converge to delta-function sections supported on the Bohr-Sommerfeld fibers. Our construction is based on a toric degeneration of flag varieties and a deformation of Kähler structure on toric varieties by symplectic potentials.
Universal embeddings of flag manifolds and rigidity phenomena
本研究证明旗流形可全纯等距嵌入不可约经典旗流形，借此建立齐性凯勒流形的新刚性现象，推广Loi-Mossa刚性定理并统一已有相关结果。
Reducible Holonomy in Closed Torsion Geometries
本文探讨了具有闭合反对称扭量和可约旋量的连接，展示其可局部定义的黎曼子流形及其基底投影几何。重新阐述了非凯勒毕舍特赫尔米特爱因斯坦流形和sHKT结构的子流形结果，并研究了半单李群上的homogeneous SKT结构的旋量分解。
Rigidity properties of holomorphic isometries into homogeneous Kähler manifolds
本文证明了齐性凯勒流形上全纯等距映射的两个刚性定理，揭示特殊旗流形、平坦空间与齐性有界域在凯勒乘积下的非相对性，并推广了多项已有结果。
Let $U/K_Θ$ be a generalized flag manifold, where $K_Θ$ is the centralizer of a torus in $U$. We study $U$-invariant almost Hermitian structures on $U/K_Θ$. The classification of these structures are naturally related with the system $R_t$ of t-roots associated to $U/K_Θ$. We introduced the notion of connectedness by triples zero sum in a general set of linear functional and proved that t-roots are connected by triples zero sum. Using this property, the invariant G1 structures on $U/K_Θ$ are completely classified. We also study the Kähler form and classified the invariant quasi Kähler structur…
The coadjoint orbits of compact Lie groups each carry a canonical (positive definite) Kähler structure, famously used to realize the group's irreducible representations in holomorphic sections of appropriate line bundles (Borel-Weil theorem). Less studied are the (indefinite) invariant *pseudo*-Kähler structures they also admit, which can be used to realize the same representations in higher cohomology of the sections (Bott's theorem). Using ``eigenflag'' embeddings, we give a very explicit description of these metrics in the case of the unitary group. As a byproduct we show that $U_n/(U_{n_1}…
In this paper we study invariant almost Hermitian geometry on generalized flag manifolds. We will focus on providing examples of Kähler like scalar curvature metric, that is, almost Hermitian structures $(g,J)$ satisfying $s=2s_{\rm C}$, where $s$ is Riemannian scalar curvature and $s_{\rm C}$ is the Chern scalar curvature.
In this paper we study invariant almost Hermitian geometry on generalized flag manifolds which the isotropy representation decompose into two or three irreducible components. We will provide a classification of such flag manifolds admitting Kähler like scalar curvature metric, that is, almost Hermitian structures $(g,J)$ satisfying $s=2s_C$ where $s$ is Riemannian scalar curvature and $s_C$ is the Chern scalar curvature.
We find the precise number of non-Kähler $Sp(n)$-invariant Einstein metrics on the generalized flag manifold $M=Sp(n)/(U(p)\times U(n-p))$ with $n\geq 3$ and $1\leq p\leq n-1$. We use an analysis on parametric systems of polynomial equations and we give some insight towards the study of such systems.
In the present paper we provide a description of complete Calabi-Yau metrics on the canonical bundle of generalized complex flag manifolds. By means of Lie theory we give an explicit description of complete Ricci-flat Kähler metrics obtained through the Calabi ansatz technique. We use this approach to provide several explicit examples of noncompact complete Calabi-Yau manifolds, these examples include canonical bundles of non-toric flag manifolds (e.g. Grassmann manifolds and full flag manifolds).
Riemannian $λ_1$-extremal Metrics on Generalized Flag Manifolds
本文利用李理论数据给出的充要条件，研究广义旗流形上 Kähler-Einstein 度量作为 Laplace 算子第一正特征值泛函临界点的 $λ_1$-极值性，并得到偏旗流形的新分类结果。
We construct the Einstein equation for an invariant Riemannian metric on the exceptional full flag manifold $M=G_2/T$. By computing a Gröbner basis for a system of polynomials of multi-variables we prove that this manifold admits exactly two non-Kähler invariant Einstein metrics. Thus $G_2/T$ turns out to be the first known example of an exceptional full flag manifold which admits at least one non-Kähler and not normal homogeneous Einstein metric.
We classify invariant almost complex structures on homogeneous manifolds of dimension 6 with semi-simple isotropy. Those with non-degenerate Nijenhuis tensor have the automorphism group of dimension either 14 or 9. An invariant almost complex structure with semi-simple isotropy is necessarily either of specified 6 homogeneous types or a left-invariant structure on a Lie group. For integrable invariant almost complex structures we classify all compatible invariant Hermitian structures on these homogeneous manifolds, indicate their integrability properties (Kahler, SNK, SKT) and mark the other i…
The classification of homogeneous Einstein metrics on flag manifolds with $b_2(M)=1$
本文针对第二贝蒂数为1的广义旗流形中两个未分类的例外E8型旗流形，用黎曼淹没方法构建爱因斯坦方程并完成齐性爱因斯坦度量分类，最终证明这类流形的非凯勒不变爱因斯坦度量数量有限。
The parallel Bismut torsion condition and the isotropy representation of flag manifolds
本文分类了大多数旗流形上具有平行Bismut挠率的不变Hermite结构，证明在简单李群且非例外群时，该结构等价于Kähler度量或Killing/标准度量的倍数。
We discuss the complex geometry of two complex five-dimensional Kähler manifolds which are homogeneous under the exceptional Lie group $G_2$. For one of these manifolds rigidity of the complex structure among all Kählerian complex structures was proved by Brieskorn, for the other one we prove it here. We relate the Kähler assumption in Brieskorn's theorem to the question of existence of a complex structure on the six-dimensional sphere, and we compute the Chern numbers of all $G_2$-invariant almost complex structures on these manifolds.
We use the procedure of reduction of Courant algebroids to reduce strong KT, hyper KT and generalized Kaehler structures on Courant algebroids. This allows us to recover results from the literature as well as explain from a different angle some of the features observed there in. As an example, we prove that the moduli space of instantons of a bundle over a SKT/HKT/generalized Kähler manifold is endowed with the same type of structure as the original manifold.
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.
We calculate the Chern classes and Chern numbers for the natural almost Hermitian structures of the partial flag manifolds F_n=SU(n+2)/S(U(n)\times U(1)\times U(1)). For all n>1 there are two invariant complex algebraic structures, which arise from the projectivizations of the holomorphic tangent and cotangent bundles of complex projective spaces. The projectivization of the cotangent bundle is the twistor space of a Grassmannian considered as a quaternionic Kähler manifold. There is also an invariant nearly Kähler structure, because F_n is a 3-symmetric space. We explain the relations between…
For a Kähler Manifold $M$, the "symplectic Dolbeault operators" are defined using the symplectic spinors and associated Dirac operators, in complete analogy to how the usual Dolbeault operators, $\bar\partial$ and $\bar\partial^*$, arise from Dirac operators on the canonical complex spinors on $M$. We give special attention to two special classes of Kähler manifolds: Riemann surfaces and flag manifolds ($G/T$ for $G$ a simply-connected compact semisimple Lie group and $T$ a maximal torus). In the case of flag manifolds, we work with the Hermitian structure induced by the Killing form and a cho…
Complete description of invariant Einstein metrics on the generalized flag manifold $SO(2n)/U(p)\times U(n-p)$
该研究通过分析参数化多项式方程组，确定了特定参数范围内广义旗流形$SO(2n)/U(p)\times U(n-p)$上非Kähler的$SO(2n)$-不变Einstein度量的精确数目，并研究了其等距问题，为同类方程组研究提供了参考。
We construct explicit complete Ricci-flat metrics on the total spaces of certain vector bundles over flag manifolds of the group $SU(n)$, for all Kähler classes. These metrics are natural generalizations of the metrics of Candelas-de la Ossa on the conifold, Pando Zayas-Tseytlin on the canonical bundle over $\mathbb{CP}^1\times \mathbb{CP}^1$, as well as the metrics on canonical bundles over flag manifolds, recently constructed by van Coevering.
Noncommutative Kähler structures were recently introduced by the second author as a framework for studying noncommutative Kähler geometry on quantum homogeneous spaces. It was subsequently observed that the notion of a positive vector bundle directly generalises to this setting, as does the Kodaira vanishing theorem. In this paper, by restricting to covariant Kähler structures of irreducible type (those having an irreducible space of holomorphic one-forms) we provide simple cohomological criteria for positivity, offering a means to avoid explicit curvature calculations. These general results a…
A Hermitian metric $ω$ on a complex manifold is called SKT or pluriclosed if $dd^cω=0$. Let M be a twistor space of a compact, anti-selfdual Riemannian manifold, admitting a pluriclosed Hermitian metric. We prove that in this case M is Kähler, hence isomorphic to $\C P^3$ or a flag space. This result is obtained from rational connectedness of the twistor space, due to F. Campana. As an aside, we prove that the moduli space of rational curves on the twistor space of a K3 surface is Stein.
Invariant generalized complex geometry on maximal flag manifolds and their moduli
本文研究极大旗流形上不变广义复结构与广义Kähler结构的模空间，通过纯旋量给出模空间的另一种描述，并刻画Weyl群作用诱导的胞腔分解。
We show that tensoring the Laplace and Dolbeault-Dirac operators of a Kähler structure (with closed integral) by a negative Hermitian holomorphic module, produces operators with spectral gaps around zero. The proof is based on the recently established Akizuki-Nakano identity of a noncommutative Kähler structure. This general framework is then applied to the Heckenberger-Kolb calculi of the irreducible quantum flag manifolds, and it is shown that twisting their Dirac and Laplace operators by negative line bundles produces a spectral gap, for q sufficiently close to 1. The main technical challen…
This article is an exposition of four loosely related remarks on the geometry of Finsler manifolds with constant positive flag curvature. <p> The first remark is that there is a canonical Kahler structure on the space of geodesics of such a manifold. <p> The second remark is that there is a natural way to construct a (not necessarily complete) Finsler n-manifold of constant positive flag curvature out of a hypersurface in suitably general position in complex projective n-space. <p> The third remark is that there is a description of the Finsler metrics of constant curvature on the 2-sphere in t…
We develop a general approach to prove K-stability of Fano varieties. The new theory is used to (a) prove the existence of Kähler-Einstein metrics on all smooth Fano hypersurfaces of Fano index two, (b) to compute the stability thresholds for hypersurfaces at generalized Eckardt points and for cubic surfaces at all points, and (c) to provide a new algebraic proof of Tian's criterion for K-stability, amongst other applications.
Homogeneous Einstein metrics on generalized flag manifolds with five isotropy summands
该研究针对具有五个迷向和项的紧单李群广义旗流形，结合旗流形纤维化与黎曼淹没技术构建齐性爱因斯坦方程，通过Gröbner基分类流形并求解，得到部分流形的全部不变爱因斯坦度量及部分流形的非Kähler-Einstein度量存在性结论。
Point vortex dynamics on Kähler twistor spaces
针对仅有的两类源自 4 维流形的 Kähler 扭量空间，本文给出 $\mathbb{CP}^n$ 的 Green 函数、点涡 Hamiltonian 与运动方程，并确定旗流形上的动量映射。
Symmetric spaces as adjoint orbits and their geometries
本文通过将经典对称空间实现为对称旗流形的余切丛，研究其测地线与拉格朗日子流形，并首次构造出单连通流形上不能承载非负截面曲率度量的向量丛例子。
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