Andrei Moroianu
We show that for $n>2$ a compact locally conformally Kähler manifold $(M^{2n},g,J)$ carrying a non-trivial parallel vector field is either Vaisman, or globally conformally Kähler, determined in an explicit way by some compact Kähler manifold of dimension $2n-2$.
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We study a natural class of LCK manifolds that we call integrable LCK manifolds: those where the anti-Lee form $η$ corresponds to an integrable distribution. As an application we obtain a characterization of unimodular integrable LCK Lie algebras as Kähler Lie algebras equipped with suitable derivations.
A locally conformally Kähler (LCK) manifold is a manifold which is covered by a Kähler manifold, with the deck transform group acting by homotheties. We show that the search for LCK metrics on Oeljeklaus-Toma manifolds leads to a (yet another) variation on Kronecker's theorem on units. In turn, this implies that on Oeljeklaus-Toma manifold associated to number fields with $2t$ complex embeddings and $s$ real embeddings with $s<t$ there is no LCK metric.
Conformal vector fields on lcK manifolds
本文证明了紧致lcK流形上的任意共形向量场关于Gauduchon度量是Killing的，且在Kähler覆盖既非平坦也非hyperkähler时该向量场是全纯的。
We prove that any compact homogeneous locally conformally Kähler manifold has parallel Lee form.
Futaki invariant on lcK manifolds with potential
本文证明 Futaki 不变量在紧致具有势的局部共形 Kähler 流形上恒为零，解决了 Ornea 和 Verbitsky 提出的问题。
Compact homogeneous locally conformally Kahler manifolds are Vaisman. A new proof
本文证明了任意齐次LCK流形容许具LCK势的度量，并以此给出了紧致齐次LCK流形是Vaisman流形的新证明。
LcK structures with holomorphic Lee vector field on Vaisman-type manifolds
本研究完整刻画了Vaisman型紧复流形上带全纯Lee向量场的局部共形Kähler（lcK）结构，给出Lee向量场不相似于Vaisman结构Lee向量场的实例，并推广证明紧复流形上带势的全纯Lee向量场lcK度量存在正且沿反Lee向量场不变的势。
We prove that a compact toric locally conformally Kähler manifold which is not Kähler admits a toric Vaisman structure, a fact which was conjectured in \cite{mmp}. This is the final step leading to the classification of compact toric locally conformally Kähler manifolds started in \cite{p} and \cite{mmp}. We also show, by constructing an example, that unlike in the symplectic case, toric locally conformally symplectic manifolds are not necessarily toric locally conformally Kähler.
Let $M$ be a complex manifold and $L$ an oriented real line bundle on M equipped with a flat connection. An LCK ("locally conformally Kahler") form is a closed, positive (1,1)-form taking values in L, and an LCK manifold is one which admits an LCK form. Locally, any LCK form is expressed as an L-valued pluri-Laplacian of a function called LCK potential. We consider a manifold $M$ with an LCK form admitting a global LCK potential, and prove that M admits a global, positive LCK potential. Then M admits a holomorphic embedding to a Hopf manifold.
We investigate the relation between holomorphic torus actions on complex manifolds of LCK type and the existence of special LCK metrics. We show that if the group of biholomorphisms of such a manifold $(M,J)$ contains a non-real compact torus, then there exists a Vaisman metric on the manifold. Moreover, we show that if the group of biholomorphisms contains a compact torus whose dimension is half the real dimension of $M$, then $(M,J)$ admits an LCK metric with positive potential. As an application, we obtain a classification of manifolds of LCK type among all the manifolds having the structur…
An LCK manifold with potential is a compact quotient of a Kahler manifold $X$ equipped with a positive Kahler potential $f$, such that the monodromy group acts on $X$ by holomorphic homotheties and multiplies $f$ by a character. The LCK rank is the rank of the image of this character, considered as a function from the monodromy group to real numbers. We prove that an LCK manifold with potential can have any rank between 1 and $b_1(M)$. Moreover, LCK manifolds with proper potential (ones with rank 1) are dense. Two errata to our previous work are given in the last Section.
A recent result of M. Kourganoff states that if $D$ is a closed, reducible, non-flat, Weyl connection on a compact conformal manifold $M$, then the universal covering of $M$, endowed with the metric whose Levi-Civita covariant derivative is the pull-back of $D$, is isometric to $\mathbb{R}^q\times N$ for some irreducible, incomplete Riemannian manifold $N$. Moreover, he characterized the case where the dimension of $N$ is $2$ by showing that $M$ is then a mapping torus of some Anosov diffeomorphism of $T^{q+1}$. We show that in this case one necessarily has $q=1$ or $q=2$.
Do products of compact complex manifolds admit LCK metrics?
针对紧复流形乘积被猜想不容许 LCK 结构的问题，本文将已知紧 LCK 流形分为带势、Inoue 型和含有理曲线三类，并证明 LCK 流形与任一类中流形的乘积均无 LCK 结构。
A locally conformally Kähler (lcK) manifold is a complex manifold $(M,J)$ together with a Hermitian metric $g$ which is conformal to a Kähler metric in the neighbourhood of each point. In this paper we obtain three classification results in locally conformally Kähler geometry. The first one is the classification of conformal classes on compact manifolds containing two non-homothetic Kähler metrics. The second one is the classification of compact Einstein locally conformally Kähler manifolds. The third result is the classification of the possible (restricted) Riemannian holonomy groups of compa…
A note on locally conformally Kaehler structures and small deformations of Hopf manifolds
本文通过构造对角型Hopf流形附近的复解析族，结合lcK势在小参数复解析族中保持的已知结论，证明了所有Hopf流形都容许局部共形Kaehler结构。
Calabi-Yau locally conformally Kähler manifolds
本文研究了Calabi-Yau局部共形Kähler流形，证明此类流形必为Vaisman流形，并探讨了其与Kodaira流形的关系。
A locally conformally Kähler (LCK) manifold is a complex manifold covered by a Kähler manifold, with the covering group acting by homotheties. We show that if such a compact manifold X admits a holomorphic submersion with positive dimensional fibers at least one of which is of Kähler type, then X is globally conformally Kähler or biholomorphic, up to finite covers, to a Vaisman manifold (i.e. a mapping torus over a circle, with Sasakian fibre). As a consequence, we show that the product between a compact non-Kähler LCK and a compact Kähler manifold cannot carry a LCK metric.
Lee classes on LCK manifolds with potential
本文研究带势 LCK 流形的 Lee 类，证明其集合构成 $H^1(M,\mathbb R)$ 中的开半空间，并为 Vaisman 情形下 Tsukada 的结果提供新的自足证明。
We show a bijective correspondence between compact toric locally conformally symplectic manifolds which admit a compatible complex structure and pairs $(C,a)$, where $C$ is a good cone in the dual Lie algebra of the torus and $a$ is a positive real number. Moreover, we prove that any toric locally conformally Kähler metric on a compact manifold admits a positive potential.
A locally conformally Kahler (LCK) manifold is a complex manifold with a Kahler structure on its covering and the deck transform group acting on it by holomorphic homotheties. One could think of an LCK manifold as of a complex manifold with a Kahler form taking values in a local system $L$, called the conformal weight bundle. The $L$-valued cohomology of $M$ is called Morse-Novikov cohomology. It was conjectured that (just as it happens for Kahler manifolds) the Morse-Novikov complex satisfies the $dd^c$-lemma. If true, it would have far-reaching consequences for the geometry of LCK manifolds.…
Compact balanced threefolds and LCK manifolds with constant holomorphic sectional curvature
该研究验证了$c\leq0$时紧致平衡三维簇满足埃尔米特几何中关于常陈全纯截面曲率的长期猜想，还将该猜想推广到典范度量联络情形并对连通紧致LCK流形成立。
Compact lcK manifolds with Gauduchon parallel vector fields
该研究刻画了紧局部共形Kähler流形上Gauduchon平行向量场与度量类型的关系，通过Bismut和乐群表征紧Vaisman流形，并得出复二维时∇^(1/3)-平行向量场对度量的约束结论。
An locally conformally Kahler (LCK) manifold with potential is a complex manifold with a cover which admits an automorphic Kahler potential. An LCK manifold with potential can be embedded to a Hopf manifold, if its dimension is at least 3. We give a functional-analytic proof of this result based on Riesz-Schauder theorem and Montel theorem. We give an alternative argument for complex surfaces, deducing embedding theorem from the Spherical Shell Conjecture.
Locally conformally Kaehler manifolds with potential
局部共形Kähler（LCK）流形$M$是被Kähler流形$\tilde M$覆盖的流形，其甲板变换群共形地作用在$\tilde M$上。如果$M$允许一个全纯流共形地作用在$\tilde M$上，则它称为Vaisman流形。LCK流形和Vaisman流形在小变形下都不是稳定的。我们定义了一类新的LCK-流形，称之为位势LCK流形，它在小变形下是闭的。所有的Vaisman流形都是具有位势的LCK流形。我们证明了一个具有势的LCK流形存在一个覆盖，通过增加一个点，该覆盖可以紧化为Stein簇。这是用来表明任何LCK流形M的潜力，$\dim M > 2$，可以嵌入到一个霍普夫流形，从而改善了类似的结果Vaisman。流形
We show that Oeljeklaus-Toma manifolds $X(K, U)$ where $K$ is a number field of signature $(s, t)$ such that $s\geq 1,$ $t\geq 2$ and $s\geq 2t$ admit no lck metric. Combined with the earlier results by K. Oeljeklaus - M. Toma and A. Dubickas this completely solves the problem of existence of LCK metrics on Oeljeklaus-Toma manifolds.
We prove that a compact lcK manifold with holomorphic Lee vector field is Vaisman provided that either the Lee field has constant norm or the metric is Gauduchon (i.e., the Lee field is divergence-free). We also give examples of compact lcK manifolds with holomorphic Lee vector field which are not Vaisman.
On products of locally conformally Kähler manifolds
本文证明了正维紧复流形的乘积不存在严格局部共形Kähler度量，解决了lcK几何中长期未解的猜想。
Bimeromorphic geometry of LCK manifolds
本文研究Kähler形式在最小Kähler覆盖上恰当时的LCK流形，证明其任意双亚纯映射均为全纯映射，从而具有唯一极小模型，并适用于Hopf流形、复子流形及OT流形。
Functionals for the Study of LCK Metrics on Compact Complex Manifolds
本文通过引入并研究一种随流形复维数（2或更高）而异的泛函，为紧复流形上局部共形Kähler度量的存在性问题提供了新的研究方法。
p-lck with potential structure on LVMB manifolds
本文将 lck 结果推广至 LVMB 流形，刻画带势 lck 流形的稀缺性，构造反例否定 p≥m−1 猜想，并提出在假设 (H) 下 N 为带势 1-lck 的新猜想。
Vaisman's theorem for locally conformally Kähler (lcK) compact manifolds states that any lcK metric on a compact complex manifold which admits a Kähler metric is, in fact, globally conformally Kähler (gcK). In this paper, we extend this theorem to compact complex spaces with singularities.
In this article we introduce a generalization of locally conformally Kaehler metrics from complex manifolds to complex analytic spaces with singularities and study which properties of locally conformally Kaehler manifolds still hold in this new setting. We prove that if a complex analytic space has only quotient singularities, then it admits a locally conformally Kaehler metric if and only if its universal cover admits a Kaehler metric such that the deck automorphisms act by homotheties of the Kaehler metric. We also prove that the blow-up at a point of a LCK complex space is also LCK.
A Hermitian structure on a manifold is called locally conformally Kähler (LCK) if it locally admits a conformal change which is Kähler. In this survey we review recent results of invariant LCK structures on solvmanifolds and present original results regarding the canonical bundle of solvmanifolds equipped with a Vaisman structure, that is, a LCK structure whose associated Lee form is parallel.
A manifold M is locally conformally Kahler (LCK) if it admits a Kahler covering with monodromy acting by holomorphic homotheties. For a compact connected group G acting on an LCK manifold by holomorphic automorphisms, an averaging procedure gives a G-invariant LCK metric. Suppose that U(1) acts on an LCK manifold M by holomorphic isometries, and the lifting of this action to the Kahler cover of M is not isometric. We show that the cover admits an automorphic Kahler potential, and hence can be embedded to a Hopf manifold.
Locally conformally Kahler (LCK) manifolds with potential are those which admit a Kahler covering with a proper, automorphic Kaehler potential. Existence of a potential can be characterized cohomologically as a vanishing of a certain cohomology class, called the Bott-Chern class. Compact LCK manifolds with potential are stable at small deformations and admit holomorphic embeddings into Hopf manifolds. This class strictly includes the Vaisman manifolds. We show that every compact LCK manifold with potential can be deformed into a Vaisman manifold. Therefore, every such manifold is diffeomorphic…
On LCK Geometry of Gauduchon Connections
本文研究Gauduchon联络在LCK和Vaisman流形中的适用性，通过相关问题与条件确定Bismut联络是最优选择，并研究其在Hopf流形上的对应方程与结构刻画。
A locally conformally Kahler (LCK) manifold is a manifold which is covered by a Kahler manifold, with the deck transform group acting by homotheties. We show that the blow-up of a compact LCK manifold along a complex submanifold admits an LCK structure if and only if this submanifold is globally conformally Kahler. We also prove that a twistor space (of a compact 4-manifold, a quaternion-Kahler manifold or a Riemannian m anifold) cannot admit an LCK metric, unless it is Kahler.
Algebraic cones of LCK manifolds with potential
本文研究可嵌入 Hopf 流形的带位势 LCK 流形，证明其 $\Z$-覆盖为代数锥，且仿射代数结构不依赖原流形选择，并给出等价的内蕴定义。
For elliptic principal bundles $π:X\ra B$ over Kähler manifolds it was shown by Blanchard that $X$ has a Kähler metric if and only both Chern classes (with real coefficients) of $π$ vanish. For some elliptic principal bundles, when the span of these Chern classes is 1-dimensional, it was shown by Vaisman that $X$ carry locally conformally Kähler (LCK, for short) metrics. We show that in the case when the Chern classes are linearly independent, $X$ carries no LCK metric.
Locally Conformally Kähler Manifolds of Algebraic Codimension One
研究代数余维数为1的紧致局部共形Kähler流形，证明其双亚纯于等平凡椭圆纤维化全空间，且存在一个变更双亚纯支配具有椭圆曲线自由作用的流形。
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。
Morse-Novikov cohomology of locally conformally Kähler manifolds
局部共形Kahler（LCK）流形是一个具有Kahler覆盖的复流形，其中单值映射通过同素映射作用于该覆盖上。我们定义了三个上同调不变量，李类，Morse-Novikov类和Bott-Chern类，LCK结构。这些不变量一起扮演着与Kahler几何中的Kahler类相同的角色。如果两个LCK结构的这些类重合，则这些结构之间的差异可以由光滑势表示，类似于Kahler的情况。我们证明了Vaisman流形的Morse-Novikov类和Bott-Chern类为零。此外，对于Vaisman流形上的任何LCK-结构，我们证明了它的Morse-Novikov类为零。证明了Bott-Chern类为零的紧致LCK-流形$M$在Hopf流形上有一个全纯嵌入，如果$\dim_\C M \geq 3$，这一结果与Kodaira嵌入定理相似.
An LCK manifold with potential is a compact quotient M of a Kahler manifold X equipped with a positive plurisubharmonic function f, such that the monodromy group acts on $X$ by holomorphic homotheties and maps f to a function proportional to f. It is known that M admits an LCK potential if and only if it can be holomorphically embedded to a Hopf manifold. We prove that any non-Vaisman LCK manifold with potential contains a complex surface with normalization biholomorphic to a Hopf surface H. Moreover, H can be chosen non-diagonal, hence, also not admitting a Vaisman structure.
A locally conformally Kahler (LCK) manifold is a complex manifold admitting a Kahler covering M, such that its monodromy acts on this covering by homotheties. A compact LCK manifold is called LCK with potential if M admits an authomorphic Kahler potential. It is known that in this case it is an algebraic cone, that is, the set of all non-zero vectors in the total space of an anti-ample line bundle over a projective orbifold. We start with an algebraic cone C, and show that the set of Kahler metrics with potential which could arise from an LCK structure is in bijective correspondence with the s…
Oeljeklaus-Toma (OT) manifolds are complex non-Kähler manifolds whose construction arises from specific number fields. In this note, we compute their de Rham cohomology in terms of invariants associated to the background number field. This is done by two distinct approaches, one using invariant cohomology and the other one using the Leray-Serre spectral sequence. In addition, we compute also their Morse-Novikov cohomology. As an application, we show that the low degree Chern classes of any complex vector bundle on an OT manifold vanish in the real cohomology. Other applications concern the OT …
A manifold $M$ is locally conformally Kahler (LCK) if it admits a Kahler covering with monodromy acting by holomorphic homotheties. Let $M$ be an LCK manifold admitting a holomorphic conformal flow of diffeomorphisms, lifted to a non-isometric homothetic flow on its covering. We show that $M$ admits an automorphic potential. This version was added 10 years after publication to correct some errors in the original.
A locally conformally Kähler (LCK) manifold is a complex manifold whose universal cover is Kähler with monodromy group acting on the universal cover by holomorphic homotheties. A Vaisman manifold $M$ is a compact non-Kähler LCK manifold admitting an action of a holomorphic conformal flow lifting to an action on a Kähler cover by nontrivial homotheties. When the orbits of the action on $M$ are compact, it is known that every stable holomorphic vector bundle over $M$, $\dim(M) \geq 3$, is equivariant and filtrable. In the present paper we generalize this result to irregular Vaisman manifolds.
Immersion theorem for Vaisman manifolds
局部共形Kaehler（LCK）流形是一个复流形，其中M是Kaehler覆盖，M上的单值映射是通过Kaehler同素映射作用的.一个紧致LCK流形是Vaisman流形，如果它允许一个作用在M上的全纯流。我们证明了Kodaira嵌入定理的一个非Kaehler类似物：任何紧Vaisman流形都承认一个Hopf流形的自然全纯浸入。作为应用，我们得到了任何Sasakian流形对奇维球面都有接触浸入。
We study compact toric strict locally conformally Kähler manifolds. We show that the Kodaira dimension of the underlying complex manifold is $-\infty$ and that the only compact complex surfaces admitting toric strict locally conformally Kähler metrics are the diagonal Hopf surfaces. We also show that every toric Vaisman manifold has lcK rank 1 and is isomorphic to the mapping torus of an automorphism of a toric compact Sasakian manifold.
Let $M^{2n}$ be a compact Riemannian manifold of non-positive (resp. negative) sectional curvature. We call $(M,J,θ)$ a $d$(bounded) locally conformally Kähler manifold if the lifted Lee form $\tildeθ$ on the universal covering space of $M$ is $d$(bounded). We shown that if $M^{2n}$ is homeomorphic to a $d$(bounded) LCK manifold, then its Euler number satisfies the inequality $(-1)^{n}χ(M^{2n})\geq$ (resp. $>$) $0$.
Non-linear Hopf manifolds are locally conformally Kahler
本文证明由非线性全纯收缩映射定义的 Hopf 流形可全纯嵌入线性 Hopf 流形，并因此具有局部共形 Kähler 度量。
Oeljeklaus-Toma (OT) manifolds are certain compact complex manifolds built from number fields. Conversely, we show that the fundamental group often pins down the number field uniquely. We relate the first homology to some interesting ideal. OT manifolds are never Kähler, but carry an LCK metric (locally conformally Kähler). Oeljeklaus and Toma used them to disprove a conjecture on the Betti numbers of spaces carrying LCK metrics. Mysteriously, the volume for this metric gives the discriminant and Dirichlet regulator of the number field. Totally unlike the above topological aspects, there does …
A locally conformally Kahler manifold is a Hermitian manifold $(M,I,ω)$ satisfying $dω=θ\wedge ω$, where $θ$ is a closed 1-form, called the Lee form of $M$. It is called pluricanonical if $\nablaθ$ is of Hodge type $(2,0)+(0,2)$, where $\nabla$ is the Levi-Civita connection, and Vaisman if $\nablaθ=0$. We show that a compact LCK manifold is pluricanonical if and only if the Lee form has constant length and the Kahler form of its covering admits an automorphic potential. Using a degenerate Monge-Ampere equation and the classification of surfaces of Kahler rank one, due to Brunella, Chiose and T…
Principles of Locally Conformally Kahler Geometry
本书系统介绍 LCK 流形的几何原理、研究现状、习题与开放问题，面向复几何方向研究生。
We classify and investigate locally conformally Kähler structures on four-dimensional solvable Lie algebras up to linear equivalence. As an application we can produce many examples in higher dimension, here including lcK structures on Oeljeklaus-Toma manifolds, and we also give a geometric interpretation of some of the $4$-dimensional structures in our classification.
We review the properties of the Morse-Novikov cohomology and compute it for all known compact complex surfaces with locally conformally Kähler metrics. We present explicit computations for the Inoue surfaces $\mathcal{S}^0$, $\mathcal{S}^+$, $\mathcal{S}^-$ and classify the locally conformally Kähler (and the tamed locally conformally symplectic) forms on $\mathcal{S}^0$. We prove the nonexistence of LCK metrics with potential and more generally, of $d_θ$-exact LCK metrics on Inoue surfaces and Oeljeklaus-Toma manifolds.
Constant $k$th-mixed curvature on locally conformal Kähler manifolds
本文证明紧致局部共形 Kähler 流形若具有非零常数 $k$ 阶混合曲率，则其度量必为 Kähler，并给出相关恒等式与障碍。
On pluricanonical locally conformally almost Kähler metrics
研究 pluricanonical 局部 conformally 近 Kähler 度量的性质，证明其 Lee 形与度量的关系，并探讨其与辛结构的关联。
Holomorphic tensors on Vaisman manifolds
本文证明了 Vaisman 流形上的全纯张量关于 Lee 场具有不变性，据此计算了其 Kodaira 维数，并证明代数锥的 $Z$-商流形与底流形的 Kodaira 维数相等及该维数的形变稳定性。
We give the parallelism between locally conformal symplectic manifolds and contact manifolds. We also give the generalization of exact contact manifolds.
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