Nearly Kähler and Hermitian <Emphasis Type="Italic">f</Emphasis>-structures on homogeneous Φ-spaces of order <Emphasis Type="Italic">k</Emphasis> with special metrics

We consider arbitrary homogeneous Φ-spaces of order k ≥ 3 of semisimple compact Lie groups G in the case of a series of special metrics. We give formulas for the Nomizu function of the Levi-Civita connection of these metrics. Using these formulas and other relations for Φ-spaces of order k, we prove necessary and sufficient conditions for the canonical f-structures on these spaces to lie in some generalized Hermitian geometry classes of f-structures: nearly Kähler (NKf-structures) and Hermitian (Hf-structures).     

Nearly Kähler and Hermitian f-structures on homogeneous Φ-spaces of order 6

For a canonical f-structure on a naturally reductive homogeneous Φ-space of order 6, we establish necessary and sufficient conditions under which this structure is a nearly Kähler or a Hermitian f-structure.     

Invariant affinor and sub-Kähler structures on homogeneous spaces

We consider G-invariant affinor metric structures and their particular cases, sub-Kähler structures, on a homogeneous space G/H. The affinor metric structures generalize almost Kähler and almost contact metric structures to manifolds of arbitrary dimension. We consider invariant sub-Riemannian and sub-Kähler structures related to a fixed 1-form with a nontrivial radical. In addition to giving some results for homogeneous spaces of arbitrary dimension, we study these structures separately on the homogeneous spaces of dimension 4 and 5.     

Homogeneous Kähler and Hamiltonian manifolds

We consider actions of reductive complex Lie groups \({G=K^\mathbb{C}}\) on Kähler manifolds X such that the K-action is Hamiltonian and prove then that the closures of the G-orbits are complex-analytic in X. This is used to characterize reductive homogeneous Kähler manifolds in terms of their isotropy subgroups. Moreover we show that such manifolds admit K-moment maps if and only if their isotropy groups are algebraic.     

Sasakian structures on CR-manifolds

A contact manifold M can be defined as a quotient of a symplectic manifold X by a proper, free action of \(\mathbb{R}\), with the symplectic form homogeneous of degree 2. If X is also Kähler, and its metric is homogeneous of degree 2, M is called Sasakian. A Sasakian manifold is realized naturally as a level set of a Kähler potential on a complex manifold, hence it is equipped with a pseudoconvex CR-structure. We show that any Sasakian manifold M is CR-diffeomorphic to an S ¹-bundle of unit vectors in a positive line bundle on a projective Kähler orbifold. This induces an embedding of M into an algebraic cone C. We show that this embedding is uniquely defined by the CR-structure. Additionally, we classify the Sasakian metrics on an odd-dimensional sphere equipped with a standard CR-structure.     

Strongly convex weakly complex Berwald metrics and real Landsberg metrics

Under the assumption that' is a strongly convex weakly Khler Finsler metric on a complex manifold M, we prove that F is a weakly complex Berwald metric if and only if F is a real Landsberg metric.This result together with Zhong(2011) implies that among the strongly convex weakly Kahler Finsler metrics there does not exist unicorn metric in the sense of Bao(2007). We also give an explicit example of strongly convex Kahler Finsler metric which is simultaneously a complex Berwald metric, a complex Landsberg metric,a real Berwald metric, and a real Landsberg metric.     

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.     

Sur les exposants de Lyapounov des applications méromorphes

Let f be a dominating meromorphic self-map of a compact Kähler manifold. We give an inequality for the Lyapounov exponents of some ergodic measures of f using the metric entropy and the dynamical degrees of f. We deduce the hyperbolicity of some measures.     

Hermitian <Emphasis Type="Italic">f</Emphasis>-structures on 6-dimensional filiform Lie groups

We consider left-invariant f-structures on 6-dimensional filiform Lie groups and present sufficient conditions under which special left-invariant f-structures on these Lie groups belong to the class of Hermitian f-structures. We also give particular examples of the corresponding f-structures.     

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.     

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.     

The energy of a Kähler class on admissible manifolds

On a compact complex manifold of Kähler type, the energy E(Ω) of a Kähler class Ω is given by the squared L ²-norm of the projection onto the space of holomorphic potentials of the scalar curvature of any Kähler metric representing the said class, and any one such metric whose scalar curvature has squared L ²-norm equal to E(Ω) must be an extremal representative of Ω. A strongly extremal metric is an extremal metric representing a critical point of E(Ω) when restricted to the set of Kähler classes of fixed positive top cup product. We study the existence of strongly extremal metrics and critical points of E(Ω) on certain admissible manifolds, producing a number of nontrivial examples of manifolds that carry this type of metrics, and where in many of the cases, the class that they represent is one other than the first Chern class, and some examples of manifolds where these special metrics and classes do not exist. We also provide a detailed analysis of the gradient flow of E(Ω) on admissible ruled surfaces, show that this dynamical system can be extended to one beyond the Kähler cone, and analyze the convergence of solution paths at infinity in terms of conditions on the initial data, in particular proving that for any initial data in the Kähler cone, the corresponding path is defined for all t, and converges to a unique critical class of E(Ω) as time approaches infinity.     

Curvature identities for principle <Emphasis Type="Italic">T</Emphasis><Superscript>1</Superscript>-bundles over almost Hermitian manifolds

We show that the identities R ₁, R ₂ and R ₃ for an almost Hermitian structure S on the base of the canonical principal T ¹-bundle are equivalent to their contact analogs for the induced almost contact metric structure S ^# on the total space of this bundle. We prove that the canonical connection of the canonical principal T ¹-bundle over a Hermitian or a quasi-Kähler manifold of class R ₃ is normal. We also prove that that the canonical connection of the canonical principal T ¹-bundle over a Vaisman-Gray manifold M of class R ₃ is normal if and only if the Lee vector of M belongs to the center of the adjoint K-algebra.     

Cohomogeneity-Three HyperKähler Metrics on Nilpotent Orbits

Let \({\cal O}\) be a nilpotent orbit in ?^? where G is a compact, simple group and ? = Lie(G). It is known that \({\cal O}\) carries a unique G-invariant hyperKähler metric admitting a hyperKähler potential compatible with the Kirillov–Kostant–Souriau symplectic form. In this work, the hyperKähler potential is explicitly calculated when \({\cal O}\) is of cohomogeneity three under the action of G. It is found that such a structure lies on a one-parameter family of hyperKähler metrics with G-invariant Kähler potentials if and only if ? is Sp3, su6, So7, So12 or e7 and otherwise is the unique G-invariant hyperKähler metric with G-invariant Kähler potential.     

A Factorization Theorem for Curves with Vanishing Self-Intersection

Let C be a connected divisor in a compact Kähler manifold such that the self-intersection of C, computed with respect to a Kähler metric, vanishes. Assume that the normal closure of the image of \(\pi _{1}(C)\) in \(\pi _{1}(Y)\) has infinite index. Then there exists a holomorphic map f from Y to a curve B such that C is a fiber. The conclusion holds if one assumes that the image of \(\pi _{1}(C)\) is amenable but \(\pi _{1}(Y)\) is non-amenable.     

Real <Emphasis Type="Italic">K</Emphasis>3 surfaces without real points,equivariant determinant of the Laplacian,and the Borcherds Φ-function

We consider an equivariant analogue of a conjecture of Borcherds. Let (Y, σ) be a real K3 surface without real points. We shall prove that the equivariant determinant of the Laplacian of (Y, σ) with respect to a σ-invariant Ricci-flat Kähler metric is expressed as the norm of the Borcherds Φ-function at the “period point”. Here the period of (Y, σ) is not the one in algebraic geometry.     

Special Kähler–Ricci potentials and Ricci solitons

On a manifold of dimension at least six, let (g, τ) be a pair consisting of a Kähler metric g which is locally Kähler irreducible, and a nonconstant smooth function τ. Off the zero set of τ, if the metric \({\widehat{g}=g/\tau^{2}}\) is a gradient Ricci soliton which has soliton function 1/τ, we show that \({\widehat{g}}\) is Kähler with respect to another complex structure, and locally of a type first described by Koiso, and also Cao. Moreover, τ is a special Kähler–Ricci potential, a notion defined in earlier works of Derdzinski and Maschler. The result extends to dimension four with additional assumptions. We also discuss a Ricci–Hessian equation, which is a generalization of the soliton equation, and observe that the set of pairs (g, τ) satisfying a Ricci–Hessian equation is invariant, in a suitable sense, under the map \({(g,\tau) \rightarrow (\widehat{g},1/\tau)}\) .     

Compact Manifolds Covered by a Torus

Let X be a compact complex manifold which is the image of a complex torus by a holomorphic surjective map A→X. We prove that X is Kähler and that up to a finite étale cover, X is a product of projective spaces by a torus.     

Riemannian <Emphasis Type="Italic">M</Emphasis>-spaces with homogeneous geodesics

We investigate homogeneous geodesics in a class of homogeneous spaces called M-spaces, which are defined as follows. Let G / K be a generalized flag manifold with \(K=C(S)=S\times K_1\), where S is a torus in a compact simple Lie group G and \(K_1\) is the semisimple part of K. Then, the associated M-space is the homogeneous space \(G/K_1\). These spaces were introduced and studied by H. C. Wang in 1954. We prove that for various classes of M-spaces the only g.o. metric is the standard metric. For other classes of M-spaces we give either necessary, or necessary and sufficient conditions, so that a G-invariant metric on \(G/K_1\) is a g.o. metric. The analysis is based on properties of the isotropy representation \(\mathfrak {m}=\mathfrak {m}_1\oplus \cdots \oplus \mathfrak {m}_s\) of the flag manifold G / K [as \({{\mathrm{Ad}}}(K)\)-modules].