Topology of Some Kähler Manifolds II

1969 ◽  
Vol 12 (4) ◽  
pp. 457-460 ◽  
Author(s):  
K. Srinivasacharyulu
Keyword(s):  
Euler Number ◽  
Complex Surface ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Almost Complex ◽  
Homogeneous Complex ◽  
Homogeneous Complex Manifold ◽  
Simply Connected ◽  
Complex Projective ◽  
Positively Curved

Topology of positively curved compact Kähler manifolds had been studied by several authors (cf. [6; 2]); these manifolds are simply connected and their second Betti number is one [1]. We will restrict ourselves to the study of some compact homogeneous Kähler manifolds. The aim of this paper is to supplement some results in [9]. We prove, among other results, that a compact, simply connected homogeneous complex manifold whose Euler number is a prime p ≥ 2 is isomorphic to the complex projective space Pp-1 (C); in the p-1 case of surfaces, we prove that a compact, simply connected, homogeneous almost complex surface with Euler-Poincaré characteristic positive, is hermitian symmetric.

Almost isotropic Kähler manifolds

2020 ◽  
Vol 2020 (767) ◽  
pp. 1-16
Author(s):  
Benjamin Schmidt ◽  
Krishnan Shankar ◽  
Ralf Spatzier
Keyword(s):  
Classical Result ◽  
Jacobi Operator ◽  
Unit Vector ◽  
Complete Riemannian Manifold ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Jacobi Operators ◽  
Sectional Curvatures ◽  
Simply Connected ◽  
Complex Projective

AbstractLet M be a complete Riemannian manifold and suppose {p\in M}. For each unit vector {v\in T_{p}M}, the Jacobi operator, {\mathcal{J}_{v}:v^{\perp}\rightarrow v^{\perp}} is the symmetric endomorphism, {\mathcal{J}_{v}(w)=R(w,v)v}. Then p is an isotropic point if there exists a constant {\kappa_{p}\in{\mathbb{R}}} such that {\mathcal{J}_{v}=\kappa_{p}\operatorname{Id}_{v^{\perp}}} for each unit vector {v\in T_{p}M}. If all points are isotropic, then M is said to be isotropic; it is a classical result of Schur that isotropic manifolds of dimension at least 3 have constant sectional curvatures. In this paper we consider almost isotropic manifolds, i.e. manifolds having the property that for each {p\in M}, there exists a constant {\kappa_{p}\in\mathbb{R}} such that the Jacobi operators {\mathcal{J}_{v}} satisfy {\operatorname{rank}({\mathcal{J}_{v}-\kappa_{p}\operatorname{Id}_{v^{\perp}}}% )\leq 1} for each unit vector {v\in T_{p}M}. Our main theorem classifies the almost isotropic simply connected Kähler manifolds, proving that those of dimension {d=2n\geqslant 4} are either isometric to complex projective space or complex hyperbolic space or are totally geodesically foliated by leaves isometric to {{\mathbb{C}}^{n-1}}.

scholarly journals Strongly not relatives Kähler manifolds

2017 ◽  
Vol 4 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Michela Zedda
Keyword(s):  
Projective Space ◽  
Positive Constant ◽  
Complex Projective Space ◽  
Kähler Manifold ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Infinite Dimensional ◽  
Kahler Manifold ◽  
Hartogs Domains ◽  
Complex Projective

Abstract In this paper we study Kähler manifolds that are strongly not relative to any projective Kähler manifold, i.e. those Kähler manifolds that do not share a Kähler submanifold with any projective Kähler manifold even when their metric is rescaled by the multiplication by a positive constant. We prove two results which highlight some relations between this property and the existence of a full Kähler immersion into the infinite dimensional complex projective space. As application we get that the 1-parameter families of Bergman-Hartogs and Fock-Bargmann-Hartogs domains are strongly not relative to projective Kähler manifolds.

Coupled Vortex Equations on Complete Kähler Manifolds

2008 ◽  
Vol 51 (3) ◽  
pp. 467-480
Author(s):  
Yue Wang
Keyword(s):  
Symmetric Spaces ◽  
Existence Results ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Convex Domains ◽  
Hermitian Symmetric Spaces ◽  
Vortex Equations ◽  
Curved Manifolds ◽  
Pseudo Convex ◽  
Simply Connected

AbstractIn this paper, we first investigate the Dirichlet problem for coupled vortex equations. Secondly, we give existence results for solutions of the coupled vortex equations on a class of complete noncompact Kähler manifolds which include simply-connected strictly negative curved manifolds, Hermitian symmetric spaces of noncompact type and strictly pseudo-convex domains equipped with the Bergmann metric.

scholarly journals Chern Class Inequalities on Polarized Manifolds and Nef Vector Bundles

2020 ◽  
Author(s):  
Ping Li ◽  
Fangyang Zheng
Keyword(s):  
Chern Class ◽  
Vector Bundles ◽  
Euler Number ◽  
Type Inequality ◽  
Chern Number ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Bisectional Curvature ◽  
Chern Numbers ◽  
Nef Vector Bundles

Abstract This article is concerned with Chern class and Chern number inequalities on polarized manifolds and nef vector bundles. For a polarized pair $(M,L)$ with $L$ very ample, our 1st main result is a family of sharp Chern class inequalities. Among them the 1st one is a variant of a classical result and the equality case of the 2nd one is a characterization of hypersurfaces. The 2nd main result is a Chern number inequality on it, which includes a reverse Miyaoka–Yau-type inequality. The 3rd main result is that the Chern numbers of a nef vector bundle over a compact Kähler manifold are bounded below by the Euler number. As an application, we classify compact Kähler manifolds with nonnegative bisectional curvature whose Chern numbers are all positive. A conjecture related to the Euler number of compact Kähler manifolds with nonpositive bisectional curvature is proposed, which can be regarded as a complex analogue to the Hopf conjecture.

scholarly journals Proof of the radical conjecture for homogeneous Kähler manifolds

1989 ◽  
Vol 114 ◽  
pp. 77-122 ◽  
Author(s):  
Josef Dorfmeister
Keyword(s):  
Lie Algebra ◽  
Kähler Manifold ◽  
Transitive Group ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Fiber Space ◽  
Kahler Manifold ◽  
Solvable Ideal ◽  
Simply Connected ◽  
Bounded Homogeneous Domain

In 1967 Gindikin and Vinberg stated the Fundamental Conjecture for homogeneous Kähler manifolds. It (roughly) states that every homogeneous Kähler manifold is a fiber space over a bounded homogeneous domain for which the fibers are a product of a flat with a simply connected compact homogeneous Kähler manifold. This conjecture has been proven in a number of cases (see [6] for a recent survey). In particular, it holds if the homogeneous Kähler manifold admits a reductive or an arbitrary solvable transitive group of automorphisms [5]. It is thus tempting to think about the general case. It is natural to expect that lack of knowledge about the radical of a transitive group G of automorphisms of a homogeneous Kähler manifold M is the main obstruction to a proof of the Fundamental Conjecture for M. Thus it is of importance to consider the Kähler algebra generated by the radical of the Lie algebra of G. Computations in this context suggest that one rather considers Kähler algebras generated by an arbitrary solvable ideal.

scholarly journals Représentations linéaires des groupes kählériens : factorisations et conjecture de Shafarevich linéaire

2014 ◽  
Vol 151 (2) ◽  
pp. 351-376 ◽  
Author(s):  
Fréderic Campana ◽  
Benoît Claudon ◽  
Philippe Eyssidieux
Keyword(s):  
General Type ◽  
Linear Representation ◽  
Kähler Manifolds ◽  
Fundamental Groups ◽  
Kahler Manifolds ◽  
Shafarevich Conjecture ◽  
Classical Work ◽  
Projective Manifolds ◽  
Complex Projective ◽  
Holomorphic Convexity

AbstractWe extend to compact Kähler manifolds some classical results on linear representation of fundamental groups of complex projective manifolds. Our approach, based on an interversion lemma for fibrations with tori versus general type manifolds as fibers, gives a refinement of the classical work of Zuo. We extend to the Kähler case some general results on holomorphic convexity of coverings such as the linear Shafarevich conjecture.

scholarly journals The canonical spectrum of projective toric manifolds

2018 ◽  
Vol 29 (07) ◽  
pp. 1850048
Author(s):  
Mounir hajli
Keyword(s):  
Spectral Theory ◽  
Spectral Properties ◽  
Kähler Manifolds ◽  
Combinatorial Structure ◽  
Kahler Manifolds ◽  
Toric Manifold ◽  
Closed Formula ◽  
Toric Manifolds ◽  
Complex Projective

Let [Formula: see text] be a complex projective toric manifold. We associated to [Formula: see text], a positive and closed [Formula: see text]-current called the canonical toric Kähler current of [Formula: see text] denoted by [Formula: see text], and a new invariant called the canonical spectrum of [Formula: see text]. This spectrum is obtained as the set of the eigenvalues of a singular Laplacian defined by [Formula: see text] and which is described uniquely by the combinatorial structure of [Formula: see text]. The construction of this Laplacian and the study of its spectral properties are the consequence of a generalized spectral theory of Laplacians on compact Kähler manifolds that we develop in this paper.

Almost complex submanifolds of nearly Kähler manifolds

2020 ◽  
Vol 115 (3) ◽  
pp. 353-358
Author(s):  
Limiao Lin ◽  
Luc Vrancken ◽  
Anne Wijffels
Keyword(s):  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Almost Complex ◽  
Complex Submanifolds ◽  
Nearly Kähler ◽  
Nearly Kähler Manifolds
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