scholarly journals On the geometry of affine Kähler immersions

1990 ◽  
Vol 120 ◽  
pp. 205-222 ◽  
Author(s):  
Katsumi Nomizu ◽  
Ulrich Pinkall ◽  
Fabio Podestà
Keyword(s):  
Isometric Immersion ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Complex Manifolds ◽  
Affine Immersion ◽  
Holomorphic Affine ◽  
Affine Immersions ◽  
Holomorphic Affine Immersion

In this paper we extend the work on affine immersions [N-Pi]-1 to the case of affine immersions between complex manifolds and lay the foundation for the geometry of affine Kähler immersions. The notion of affine Kähler immersion extends that of a holomorphic and isometric immersion between Kähler manifolds and can be contrasted to the notion of holomorphic affine immersion which has been established in the work of Dillen, Vrancken and Verstraelen [D-V-V] and that of Abe [A].

Introduction to Kähler Manifolds

2017 ◽  
Author(s):  
Eduardo Cattani ◽  
Phillip Griffiths
Keyword(s):  
Summer School ◽  
Differential Forms ◽  
Kähler Manifolds ◽  
Theoretical Physics ◽  
Kahler Manifolds ◽  
Complex Manifolds ◽  
Linear Transformations ◽  
Harmonic Forms ◽  
Hodge Structures ◽  
Kähler Structures

This chapter provides an introduction to the basic results on the topology of compact Kähler manifolds that underlie and motivate Hodge theory. This chapter consists of five sections which correspond, roughly, to the five lectures in the course given during the Summer School at the International Centre for Theoretical Physics (ICTP). The five topics under discussion are: complex manifolds; differential forms on complex manifolds; symplectic, Hermitian, and Kähler structures; harmonic forms; and the cohomology of compact Kähler manifolds. There are also two appendices. The first collects some results on the linear algebra of complex vector spaces, Hodge structures, nilpotent linear transformations, and representations of sl(2,ℂ) and serves as an introduction to many other chapters in this volume. The second contains a new proof of the Kähler identities by reduction to the symplectic case.

scholarly journals Automorphisms and examples of compact non-Kähler manifolds

2017 ◽  
Vol 121 (1) ◽  
pp. 49
Author(s):  
Gunnar Þór Magnússon
Keyword(s):  
Kähler Manifold ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Canonical Bundle ◽  
Complex Manifolds ◽  
Kahler Manifold ◽  
Simply Connected

If $f$ is an automorphism of a compact simply connected Kähler manifold with trivial canonical bundle that fixes a Kähler class, then the order of $f$ is finite. We apply this well known result to construct compact non-Kähler manifolds. These manifolds contradict the abundance and Iitaka conjectures for complex manifolds.

Holomorphic Vanishing Theorems on Finsler Holomorphic Vector Bundles and Complex Finsler Manifolds

2018 ◽  
Vol 62 (3) ◽  
pp. 623-641
Author(s):  
Bin Shen
Keyword(s):  
Vector Fields ◽  
Vector Bundles ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Vanishing Theorems ◽  
Complex Manifolds ◽  
Finsler Manifolds ◽  
Holomorphic Vector Fields ◽  
Holomorphic Vector Bundles ◽  
Holomorphic Sections

AbstractIn this paper, we investigate the holomorphic sections of holomorphic Finsler bundles over both compact and non-compact complete complex manifolds. We also inquire into the holomorphic vector fields on compact and non-compact complete complex Finsler manifolds. We get vanishing theorems in each case according to different certain curvature conditions. This work can be considered as generalizations of the classical results on Kähler manifolds and hermitian bundles.

Isotropic and Kähler Immersions

1965 ◽  
Vol 17 ◽  
pp. 907-915 ◽  
Author(s):  
Barrett O'Neill
Keyword(s):  
Constant Curvature ◽  
Riemannian Manifolds ◽  
Isometric Immersion ◽  
Normal Curvature ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Real Constant ◽  
Cartesian Products ◽  
The Real ◽  
Holomorphic Curvature

Let Md and be Riemannian manifolds. We shall say that an isometric immersion ϕ: Md —> is isotropic provided that all its normal curvature vectors have the same length. The class of such immersions is closed under compositions and Cartesian products. Umbilic immersions (e.g. Sd ⊂ Rd+1) are isotropic, but the converse does not hold. If M and are Kähler manifolds of constant holomorphic curvature, then any Kähler immersion of M in is automatically isotropic (Lemma 6). We shall find the smallest co-dimension for which there exist non-trivial immersions of this type, and obtain similar results in the real constant-curvature case.

HARDER–NARASIMHAN FILTRATION ON NON KÄHLER MANIFOLDS

2001 ◽  
Vol 12 (05) ◽  
pp. 579-594 ◽  
Author(s):  
LAURENT BRUASSE
Keyword(s):  
Vector Bundle ◽  
Kähler Manifolds ◽  
General Context ◽  
Kahler Manifolds ◽  
Complex Manifolds ◽  
Holomorphic Bundles ◽  
Compact Complex Manifolds

In this paper we are interested in the behavior of the degree map on holomorphic bundles over compact complex manifolds. We do not assume the manifold to be Kähler and we define the degree by means of a Gauduchon metric. We show that the degree is bounded above over all coherent subsheaves of a vector bundle and that the supremum is reached. Then we prove the existence of the Harder–Narasimhan filtration in this general context.

Sunada transplantation and isogeny of intermediate Jacobians of compact Kähler manifolds

2020 ◽  
Vol 72 (1) ◽  
pp. 127-147
Author(s):  
Carolyn Gordon ◽  
Eran Makover ◽  
Bjoern Muetzel ◽  
David Webb
Keyword(s):  
Kähler Manifolds ◽  
Kahler Manifolds

Holomorphically projective mappings between generalized m-parabolic Kähler manifolds

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4865-4873 ◽  
Author(s):  
Milos Petrovic
Keyword(s):  
Linear Systems ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Covariant Derivatives ◽  
Linearly Independent ◽  
Non Linear ◽  
Non Linear Systems ◽  
Curvature Tensors ◽  
Derivatives Of

Generalized m-parabolic K?hler manifolds are defined and holomorphically projective mappings between such manifolds have been considered. Two non-linear systems of PDE?s in covariant derivatives of the first and second kind for the existence of such mappings are given. Also, relations between five linearly independent curvature tensors of generalized m-parabolic K?hler manifolds with respect to these mappings are examined.

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