On locally conformal almost Kähler manifolds

In the first section of this note, we discuss locally conformal symplectic manifolds, which are differentiable manifoldsV ²ⁿ endowed with a nondegenerate 2-form Ω such thatdΩ=θ ∧ Ω for some closed form θ. Examples and several geometric properties are obtained, especially for the case whendΩ ≠ 0 at every point. In the second section, we discuss the case when Ω above is the fundamental form of an (almost) Hermitian manifold, i.e. the case of the locally conformal (almost) Kähler manifolds. Characterizations of such manifolds are given. Particularly, the locally conformal Kähler manifolds are almost Hermitian manifolds for which some canonically associated connection (called the Weyl connection) is almost complex. Examples of locally conformal (almost) Kähler manifolds which are not globally conformal (almost) Kähler are given. One such example is provided by the well-known Hopf manifolds.     

On toric locally conformally Kähler manifolds

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.     

Locally conformal Kähler structures on compact nilmanifolds with left-invariant complex structures

Let (M, g, J) be a compact Hermitian manifold and \(\Omega\) the fundamental 2-form of (g, J). A Hermitian manifold (M, g, J) is called a locally conformal Kähler manifold if there exists a closed 1-form α such that \(d\Omega=\alpha \wedge \Omega\) . The purpose of this paper is to give a completely classification of locally conformal Kähler nilmanifolds with left-invariant complex structures.     

Transformations of locally conformally Kähler manifolds

We consider several transformation groups of a locally conformally Kähler manifold and discuss their inter-relations. Among other results, we prove that all conformal vector fields on a compact Vaisman manifold which is neither locally conformally hyperkähler nor a diagonal Hopf manifold are Killing, holomorphic and that all affine vector fields with respect to the minimal Weyl connection of a locally conformally Kähler manifold which is neither Weyl-reducible nor locally conformally hyperkähler are holomorphic and conformal.     

A note on <Emphasis Type="Italic">tt</Emphasis><Superscript>*</Superscript>-bundles over compact nearly Kähler manifolds

First, we generalize a rigidity result for harmonic maps of Gordon (Gordon (1972) Proc AM Math Soc 33: 433–437) to generalized pluriharmonic maps. We give the construction of generalized pluriharmonic maps from metric tt ^*-bundles over nearly Kähler manifolds. An application of the last two results is that any metric tt ^*-bundle over a compact nearly Kähler manifold is trivial (Theorem A). This result we apply to special Kähler manifolds to show that any compact special Kähler manifold is trivial. This is Lu’s theorem (Lu (1999) Math Ann 313: 711–713) for the case of compact special Kähler manifolds. Further we introduce harmonic bundles over nearly Kähler manifolds and study the implications of Theorem A for tt ^*-bundles coming from harmonic bundles over nearly Kähler manifolds.     

Generalized hermann theorems and conformal holomorphic submersions

It is shown that if a Kähler manifold admits a holomorphic Riemann submersion, then this manifold is locally reducible. Hermann's well-known theorems are generalized to conformal and holomorphic submersions. A method for constructing Kähler fiber spaces with holomorphic conformal (non-Riemannian) projection and totally geodesic isomorphic fibers is suggested. The method allows us to construct complete, including compact, Kähler fiber spaces of the specified type.     

Compact complex manifolds bimeromorphic to locally conformally Kähler manifolds

We study compact complex manifolds bimeromorphic to locally conformally Kähler (LCK) manifolds. This is an analogy of studying a compact complex manifold bimeromorphic to a Kähler manifold. We give a negative answer for a question of Ornea, Verbitsky, Vuletescu by showing that there exists no LCK current on blow ups along a submanifold (dim \(\ge 1\)) of Vaisman manifolds. We show that a compact complex manifold with LCK currents satisfying a certain condition can be modified to an LCK manifold. Based on this fact, we define a compact complex manifold with a modification from an LCK manifold as a locally conformally class C (LC class C) manifold. We give examples of LC class C manifolds that are not LCK manifolds. Finally, we show that all LC class C manifolds are locally conformally balanced manifolds.     

Some examples of Hermitian surfaces

The Riemannian version of the Goldberg-Sachs theorem says that a compact Einstein Hermitian surface is locally conformal Kähler. In contrast to the compact case, we show that there exists an Einstein Hermitian surface which is not locally conformal Kähler. On the other hand, it is known that on a compact Hermitian surface M ⁴, the zero scalar curvature defect implies that M ⁴ is Kähler. Contrary to the compact case, we show that there exists a non-Kähler Hermitian surface with zero scalar curvature defect.     

Kähler manifolds of quasi-constant holomorphic sectional curvatures

The Kähler manifolds of quasi-constant holomorphic sectional curvatures are introduced as Kähler manifolds with complex distribution of codimension two, whose holomorphic sectional curvature only depends on the corresponding point and the geometric angle, associated with the section. A curvature identity characterizing such manifolds is found. The biconformal group of transformations whose elements transform Kähler metrics into Kähler ones is introduced and biconformal tensor invariants are obtained. This makes it possible to classify the manifolds under consideration locally. The class of locally biconformal flat Kähler metrics is shown to be exactly the class of Kähler metrics whose potential function is only a function of the distance from the origin in ? ⁿ . Finally we show that any rotational even dimensional hypersurface carries locally a natural Kähler structure which is of quasi-constant holomorphic sectional curvatures.     

Locally conformal Kähler manifolds with potential

A locally conformally Kähler (LCK) manifold M is one which is covered by a Kähler manifold ${\widetilde M}A locally conformally K?hler (LCK) manifold M is one which is covered by a K?hler manifold [(M)\tilde]{\widetilde M} with the deck transformation group acting conformally on [(M)\tilde]{\widetilde M}. If M admits a holomorphic flow, acting on [(M)\tilde]{\widetilde M} conformally, it is called a Vaisman manifold. Neither the class of LCK manifolds nor that of Vaisman manifolds is stable under small deformations. We define a new class of LCK-manifolds, called LCK manifolds with potential, which is closed under small deformations. All Vaisman manifolds are LCK with potential. We show that an LCK-manifold with potential admits a covering which can be compactified to a Stein variety by adding one point. This is used to show that any LCK manifold M with potential, dim M ≥ 3, can be embedded into a Hopf manifold, thus improving similar results for Vaisman manifolds Ornea and Verbitsky (Math Ann 332:121–143, 2005).     

Hodge structures on cohomology algebras and geometry

We study restrictions on cohomology algebras of compact Kähler manifolds, imposed by the presence of a polarized Hodge structure on cohomology groups, compatible with the cup-product, but not depending on the h ^p,q numbers or the symplectic structure. To illustrate the effectiveness of these restrictions, we give a number of examples of compact symplectic manifolds satisfying the formality condition, the Lefschetz property and having commutative or trivial π ₁, but not having the cohomology algebra of a compact Kaehler manifold. We also prove a stability theorem for these restrictions : if a compact Kähler manifold is homeomorphic to a product X × Y, with one summand satisfying b ₁ = 0, then the cohomology algebra of each summand carries a polarized Hodge structure.     

Compact Kähler manifolds with nonpositive bisectional curvature

Let (M ⁿ , g) be a compact Kähler manifold with nonpositive bisectional curvature. We show that a finite cover is biholomorphic and isometric to a flat torus bundle over a compact Kähler manifold N ^k with c ₁ <  0. This confirms a conjecture of Yau. As a corollary, for any compact Kähler manifold with nonpositive bisectional curvature, the Kodaira dimension is equal to the maximal rank of the Ricci tensor. We also prove a global splitting result under the assumption of certain immersed complex submanifolds.     

Traceless component of the conformal curvature tensor in Kähler manifold

We investigate the traceless component of the conformal curvature tensor defined by (2.1) in Kähler manifolds of dimension ? 4, and show that the traceless component is invariant under concircular change. In particular, we determine Kähler manifolds with vanishing traceless component and improve some theorems (for example, [4, pp. 313–317]) concerning the conformal curvature tensor and the spectrum of the Laplacian acting on p (0 ? p ? 2)-forms on the manifold by using the traceless component.     

Quaternionic Kähler and Spin(7) metrics arising from quaternionic contact Einstein structures

We construct left-invariant quaternionic contact (qc) structures on Lie groups with zero and non-zero torsion and with non-vanishing quaternionic contact conformal curvature tensor, thus showing the existence of non-flat quaternionic contact manifolds. We prove that the product of the real line with a seven-dimensional manifold, equipped with a certain qc structure, has a quaternionic Kähler metric as well as a metric with holonomy contained in Spin(7). As a consequence, we determine explicit quaternionic Kähler metrics and Spin(7)-holonomy metrics, which seem to be new. Moreover, we give explicit non-compact eight dimensional almost quaternion hermitian manifolds that are not quaternionic Kähler with either a closed fundamental four form or fundamental two forms defining a differential ideal.     

Einstein almost cokähler manifolds

We study an odd‐dimensional analogue of the Goldberg conjecture for compact Einstein almost Kähler manifolds. We give an explicit non‐compact example of an Einstein almost cokähler manifold that is not cokähler. We prove that compact Einstein almost cokähler manifolds with nonnegative *‐scalar curvature are cokähler (indeed, transversely Calabi–Yau); more generally, we give a lower and upper bound for the *‐scalar curvature in the case that the structure is not cokähler. We prove similar bounds for almost Kähler Einstein manifolds that are not Kähler.     

An integral invariant from the view point of locally conformally Kähler geometry

In this article we study an integral invariant which obstructs the existence on a compact complex manifold of a volume form with the determinant of its Ricci form proportional to itself, in particular obstructs the existence of a Kähler-Einstein metric, and has been studied since 1980s. We study this invariant from the view point of locally conformally Kähler geometry. We first see that we can define an integral invariant for coverings of compact complex manifolds with automorphic volume forms. This situation typically occurs for locally conformally Kähler manifolds. Secondly, we see that this invariant coincides with the former one. We also show that the invariant vanishes for any compact Vaisman manifold.     

Poincaré polynomial and {SL(2, \mathbb{C})} -representation on Kähler manifolds

The cohomology ring of any compact Kähler manifold gives rise to an ${SL(2, \mathbb{C})}$ -representation. In this short note, we show that the character of this representation essentially is the Poincaré polynomial of this Kähler manifold, which gives a natural interpretation of the Poincaré polynomial for Kähler manifolds. Our result is an analogue to an interpretation of the χ _y genus for holomorphic symplectic manifolds due to George Thompson.     

Holomorphic submersions of locally conformally Kähler manifolds

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 small deformation of a Vaisman manifold (i.e., a mapping torus over a circle, with Sasakian fiber). As a consequence, we show that the product of a compact non-Kähler LCK and a compact Kähler manifold cannot carry a LCK metric.     

Decomposition and minimality of lagrangian submanifolds in nearly Kähler manifolds

We show that Lagrangian submanifolds in six-dimensional nearly Kähler (non-Kähler) manifolds and in twistor spaces Z ⁴ⁿ⁺² over quaternionic Kähler manifolds Q ⁴ⁿ are minimal. Moreover, we prove that any Lagrangian submanifold L in a nearly Kähler manifold M splits into a product of two Lagrangian submanifolds for which one factor is Lagrangian in the strict nearly Kähler part of M and the other factor is Lagrangian in the Kähler part of M. Using this splitting theorem, we then describe Lagrangian submanifolds in nearly Kähler manifolds of dimensions six, eight, and ten.