scholarly journals The General Definition of the Complex Monge–Ampère Operator on Compact Kähler Manifolds

2010 ◽  
Vol 62 (1) ◽  
pp. 218-239 ◽  
Author(s):  
Yang Xing
Keyword(s):  
Convergence Theorem ◽  
Convex Cone ◽  
Kähler Manifold ◽  
Kähler Manifolds ◽  
General Definition ◽  
Kahler Manifolds ◽  
Plurisubharmonic Functions ◽  
Definition Of ◽  
Compact Kähler Manifold ◽  
Monge Ampere

AbstractWe introduce a wide subclass of quasi-plurisubharmonic functions in a compact Kähler manifold, on which the complex Monge-Ampère operator is well defined and the convergence theorem is valid. We also prove that is a convex cone and includes all quasi-plurisubharmonic functions that are in the Cegrell class.

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 Zeros of Holomorphic One-Forms and Topology of Kähler Manifolds

2020 ◽  
Author(s):  
Stefan Schreieder
Keyword(s):  
Kähler Manifold ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Joint Paper ◽  
And Topology ◽  
Kahler Manifold ◽  
Compact Kähler Manifold

Abstract A conjecture of Kotschick predicts that a compact Kähler manifold $X$ fibres smoothly over the circle if and only if it admits a holomorphic one-form without zeros. In this paper we develop an approach to this conjecture and verify it in dimension two. In a joint paper with Hao [ 10], we use our approach to prove Kotschick’s conjecture for smooth projective three-folds.

scholarly journals Numerical dimension and a Kawamata–Viehweg–Nadel-type vanishing theorem on compact Kähler manifolds

2014 ◽  
Vol 150 (11) ◽  
pp. 1869-1902 ◽  
Author(s):  
Junyan Cao
Keyword(s):  
Line Bundle ◽  
Kähler Manifold ◽  
Kähler Manifolds ◽  
Line Bundles ◽  
Kahler Manifolds ◽  
Vanishing Theorem ◽  
Kahler Manifold ◽  
Singular Metrics ◽  
Compact Kähler Manifold

AbstractLet $X$ be a compact Kähler manifold and let $(L,{\it\varphi})$ be a pseudo-effective line bundle on $X$. We first define a notion of numerical dimension for pseudo-effective line bundles with singular metrics, and then discuss the properties of this numerical dimension. Finally, we prove a very general Kawamata–Viehweg–Nadel-type vanishing theorem on an arbitrary compact Kähler manifold.

scholarly journals A remark on covering of compact Kähler manifolds and applications

2021 ◽  
Vol 73 (1) ◽  
pp. 138-148
Author(s):  
V. V. Hung ◽  
H. N. Quy
Keyword(s):  
Upper Bound ◽  
Kähler Manifold ◽  
Kahler Manifolds ◽  
Hölder Continuous ◽  
Kahler Manifold ◽  
Right Hand ◽  
Regularization Techniques ◽  
The Right ◽  
Compact Kähler Manifold ◽  
Monge Ampere

UDC 517.9 Recently, Kolodziej proved that, on a compact Kähler manifold the solutions to the complex Monge – Ampére equation with the right-hand side in are Hölder continuous with the exponent depending on and (see [Math. Ann., <strong>342</strong>, 379-386 (2008)]).Then, by the regularization techniques in[J. Algebraic Geom., <strong>1</strong>, 361-409 (1992)], the authors in [J. Eur. Math. Soc., <strong>16</strong>, 619-647 (2014)] have found the optimal exponent of the solutions.In this paper, we construct a cover of the compact Kähler manifold which only depends on curvature of Then, as an application, base on the arguments in[Math. Ann., <strong>342</strong>, 379-386 (2008)], we show that the solutions are Hölder continuous with the exponent just depending on the function in the right-hand side and upper bound of curvature of  

scholarly journals Holomorphic one-forms, fibrations over the circle, and characteristic numbers of Kähler manifolds

2021 ◽  
pp. 1-9
Author(s):  
D. KOTSCHICK
Keyword(s):  
Kähler Manifold ◽  
Analogous Problem ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Smooth Manifolds ◽  
Characteristic Numbers ◽  
Kahler Manifold ◽  
Chern Numbers ◽  
Compact Kähler Manifold

Abstract We prove that the only relation imposed on the Hodge and Chern numbers of a compact Kähler manifold by the existence of a nowhere zero holomorphic one-form is the vanishing of the Hirzebruch genus. We also treat the analogous problem for nowhere zero closed one-forms on smooth manifolds.

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 Domains of definition of Monge-Ampère operators on compact Kähler manifolds

2007 ◽  
Vol 259 (2) ◽  
pp. 393-418 ◽  
Author(s):  
Dan Coman ◽  
Vincent Guedj ◽  
Ahmed Zeriahi
Keyword(s):  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Definition Of ◽  
Monge Ampere

Global and local definition of the Monge–Ampère operator on compact Kähler manifolds

2012 ◽  
Vol 350 (3-4) ◽  
pp. 153-156 ◽  
Author(s):  
Le Mau Hai ◽  
Pham Hoang Hiep ◽  
Nguyen Van Phu
Keyword(s):  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Definition Of ◽  
Global And Local ◽  
Monge Ampere

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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