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.

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 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.

The fundamental group, rational connectedness and the positivity of Kähler manifolds

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Lei Ni
Keyword(s):  
Kähler Manifold ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Rigidity Result ◽  
Complex Dimension ◽  
Kahler Manifold ◽  
Classical Work ◽  
Simply Connected ◽  
Compact Kähler Manifold

AbstractFirstly, we confirm a conjecture asserting that any compact Kähler manifold N with {\operatorname{Ric}^{\perp}>0} must be simply-connected by applying a new viscosity consideration to Whitney’s comass of {(p,0)}-forms. Secondly we prove the projectivity and the rational connectedness of a Kähler manifold of complex dimension n under the condition {\operatorname{Ric}_{k}>0} (for some {k\in\{1,\dots,n\}}, with {\operatorname{Ric}_{n}} being the Ricci curvature), generalizing a well-known result of Campana, and independently of Kollár, Miyaoka and Mori, for the Fano manifolds. The proof utilizes both the above comass consideration and a second variation consideration of [L. Ni and F. Zheng, Positivity and Kodaira embedding theorem, preprint 2020, https://arxiv.org/abs/1804.09696]. Thirdly, motivated by {\operatorname{Ric}^{\perp}} and the classical work of Calabi and Vesentini [E. Calabi and E. Vesentini, On compact, locally symmetric Kähler manifolds, Ann. of Math. (2) 71 1960, 472–507], we propose two new curvature notions. The cohomology vanishing {H^{q}(N,T^{\prime}N)=\{0\}} for any {1\leq q\leq n} and a deformation rigidity result are obtained under these new curvature conditions. In particular, they are verified for all classical Kähler C-spaces with {b_{2}=1}. The new conditions provide viable candidates for a curvature characterization of homogeneous Kähler manifolds related to a generalized Hartshone conjecture.

scholarly journals EXTREMAL ALMOST-KÄHLER METRICS

2010 ◽  
Vol 21 (12) ◽  
pp. 1639-1662 ◽  
Author(s):  
MEHDI LEJMI
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Explicit Formula ◽  
Cohomology Class ◽  
Symplectic Form ◽  
Kähler Metrics ◽  
Kahler Metrics ◽  
Integral Cohomology ◽  
Almost Kähler ◽  
Compact Kähler Manifold

We generalize the notions of the Futaki invariant and extremal vector field of a compact Kähler manifold to the general almost-Kähler case and show the periodicity of the extremal vector field when the symplectic form represents an integral cohomology class modulo torsion. We also give an explicit formula of the Hermitian scalar curvature in Darboux coordinates which allows us to obtain examples of non-integrable extremal almost-Kähler metrics saturating LeBrun's estimates.

A POSITIVITY PROPERTY FOR FOLIATIONS ON COMPACT KÄHLER MANIFOLDS

2006 ◽  
Vol 17 (01) ◽  
pp. 35-43 ◽  
Author(s):  
MARCO BRUNELLA
Keyword(s):  
Kähler Manifold ◽  
Rational Curves ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Canonical Bundle ◽  
Positivity Property ◽  
Kahler Manifold ◽  
Compact Kähler Manifold

We prove that the canonical bundle of a foliation by curves on a compact Kähler manifold is pseudoeffective, unless the foliation is a (special) foliation by rational curves.

scholarly journals On some compact almost Kähler locally symmetric space

1998 ◽  
Vol 21 (1) ◽  
pp. 69-72 ◽  
Author(s):  
Takashi Oguro
Keyword(s):  
Symmetric Space ◽  
Scalar Curvature ◽  
Kähler Manifold ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Locally Symmetric Space ◽  
Kahler Manifold ◽  
Almost Kähler

In the framework of studying the integrability of almost Kähler manifolds, we prove that if a compact almost Kähler locally symmetric spaceMis a weakly ,∗-Einstein vnanifold with non-negative ,∗-scalar curvature, thenMis a Kähler manifold.

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