scholarly journals Astheno–Kähler and Balanced Structures on Fibrations

2017 ◽  
Vol 2019 (22) ◽  
pp. 7093-7117 ◽  
Author(s):  
Anna Fino ◽  
Gueo Grantcharov ◽  
Luigi Vezzoni
Keyword(s):  
Homogeneous Spaces ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Kähler Metrics ◽  
Hermitian Metrics ◽  
Kahler Metrics ◽  
Balanced Metrics ◽  
Invariant Volume ◽  
Balanced Metric ◽  
First Chern Class

Abstract We study the existence of three classes of Hermitian metrics on certain types of compact complex manifolds. More precisely, we consider balanced, strong Kähler with torsion (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 a construction of astheno-Kähler structures on torus bundles over Kähler manifolds leading to new examples. In particular, we find examples of compact complex non-Kähler manifolds which admit a balanced and an astheno-Kähler metric, thus answering to a question in [52] (see also [24]). One of these examples is simply connected. We also show that the Lie groups SU(3) and G2 admit SKT and astheno-Kähler metrics, which are different. Furthermore, we investigate the existence of balanced metrics on compact complex homogeneous spaces with an invariant volume form, showing in particular that if a compact complex homogeneous space M with invariant volume admits a balanced metric, then its first Chern class c1(M) does not vanish. Finally we characterize Wang C-spaces admitting SKT metrics.

scholarly journals Quaternionic Kähler metrics associated with special Kähler manifolds

2015 ◽  
Vol 92 ◽  
pp. 271-287 ◽  
Author(s):  
D.V. Alekseevsky ◽  
V. Cortés ◽  
M. Dyckmanns ◽  
T. Mohaupt
Keyword(s):  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Kähler Metrics ◽  
Kahler Metrics ◽  
Special Kähler Manifolds ◽  
Quaternionic Kähler

scholarly journals A SPLITTING THEOREM FOR KÄHLER MANIFOLDS WHOSE RICCI TENSORS HAVE CONSTANT EIGENVALUES

2001 ◽  
Vol 12 (07) ◽  
pp. 769-789 ◽  
Author(s):  
VESTISLAV APOSTOLOV ◽  
TEDI DRĂGHICI ◽  
ANDREI MOROIANU
Keyword(s):  
Ricci Tensor ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Kähler Metrics ◽  
Kahler Metrics ◽  
Complex Dimension ◽  
Negative Eigenvalues ◽  
Kähler Surfaces ◽  
Almost Kähler ◽  
Compact Kähler Manifold

It is proved that a compact Kähler manifold whose Ricci tensor has two distinct constant non-negative eigenvalues is locally the product of two Kähler–Einstein manifolds. A stronger result is established for the case of Kähler surfaces. Without the compactness assumption, irreducible Kähler manifolds with Ricci tensor having two distinct constant eigenvalues are shown to exist in various situations: there are homogeneous examples of any complex dimension n ≥ 2 with one eigenvalue negative and the other one positive or zero; there are homogeneous examples of any complex dimension n ≥ 3 with two negative eigenvalues; there are non-homogeneous examples of complex dimension 2 with one of the eigenvalues zero. The problem of existence of Kähler metrics whose Ricci tensor has two distinct constant eigenvalues is related to the celebrated (still open) conjecture of Goldberg [24]. Consequently, the irreducible homogeneous examples with negative eigenvalues give rise to complete Einstein strictly almost Kähler metrics of any even real dimension greater than 4.

On the Kähler-likeness on nearly Kähler manifolds

2020 ◽  
pp. 2150132
Author(s):  
Masaya Kawamura
Keyword(s):  
Kähler Manifold ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Hermitian Metrics ◽  
Kahler Manifold ◽  
Nearly Kähler ◽  
Nearly Kähler Manifolds ◽  
Nearly Kähler Manifold

We introduce Kähler-like, G-Kähler-like almost Hermitian metrics. We characterize the Kähler-likeness and the G-Kähler-likeness, and show that these properties are equivalent on nearly Kähler manifolds. Furthermore, we prove that a nearly Kähler manifold with the Kähler-likeness is Kähler.

scholarly journals KÄHLER GEOMETRY OF TORIC VARIETIES AND EXTREMAL METRICS

1998 ◽  
Vol 09 (06) ◽  
pp. 641-651 ◽  
Author(s):  
MIGUEL ABREU
Keyword(s):  
Scalar Curvature ◽  
Convex Polytopes ◽  
Constant Scalar Curvature ◽  
Affine Function ◽  
Combinatorial Identity ◽  
Kähler Metrics ◽  
Kahler Metrics ◽  
Total Integral ◽  
First Chern Class ◽  
Moment Polytope

A (symplectic) toric variety X, of real dimension 2n, is completely determined by its moment polytope Δ ⊂ ℝn. Recently Guillemin gave an explicit combinatorial way of constructing "toric" Kähler metrics on X, using only data on Δ. In this paper, differential geometric properties of these metrics are investigated using Guillemin's construction. In particular, a nice combinatorial formula for the scalar curvature R is given, and the Euler–Lagrange condition for such "toric" metrics being extremal (in the sense of Calabi) is proven to be R being an affine function on Δ ⊂ ℝn. A construction, due to Calabi, of a 1-parameter family of extremal Kähler metrics of non-constant scalar curvature on [Formula: see text] is recast very simply and explicitly using Guillemin's approach. Finally, we present a curious combinatorial identity for convex polytopes Δ ⊂ ℝn that follows from the well-known relation between the total integral of the scalar curvature of a Kähler metric and the wedge product of the first Chern class of the underlying complex manifold with a suitable power of the Kähler class.

scholarly journals Cohomologically Kähler manifolds with no Kähler metrics

2003 ◽  
Vol 2003 (52) ◽  
pp. 3315-3325 ◽  
Author(s):  
Marisa Fernández ◽  
Vicente Muñoz ◽  
José A. Santisteban
Keyword(s):  
Fundamental Group ◽  
Kähler Manifold ◽  
Fundamental Groups ◽  
Kahler Manifolds ◽  
Kähler Metrics ◽  
Kahler Metrics ◽  
Lefschetz Property ◽  
Open Question ◽  
Kahler Metric ◽  
Compact Kähler Manifold

We show some examples of compact symplectic solvmanifolds, of dimension greater than four, which are cohomologically Kähler and do not admit Kähler metric since their fundamental groups cannot be the fundamental group of any compact Kähler manifold. Some of the examples that we study were considered by Benson and Gordon (1990). However, whether such manifolds have Kähler metrics was an open question. The formality and the hard Lefschetz property are studied for the symplectic submanifolds constructed by Auroux (1997) and some consequences are discussed.

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