scholarly journals Geometry and Automorphisms of Non-Kähler Holomorphic Symplectic Manifolds

2021 ◽  
Author(s):  
Fedor Bogomolov ◽  
Nikon Kurnosov ◽  
Alexandra Kuznetsova ◽  
Egor Yasinsky
Keyword(s):  
Automorphism Group ◽  
Projective Space ◽  
Kähler Manifold ◽  
Symplectic Manifolds ◽  
Finite Degree ◽  
Partial Classification ◽  
Kahler Manifold ◽  
Degeneracy Locus ◽  
Algebraic Reduction

Abstract We consider the only one known class of non-Kähler irreducible holomorphic symplectic manifolds, described in the works by D. Guan and the 1st author. Any such manifold $Q$ of dimension $2n-2$ is obtained as a finite degree $n^2$ cover of some non-Kähler manifold $W_F$, which we call the base of $Q$. We show that the algebraic reduction of $Q$ and its base is the projective space of dimension $n-1$. Besides, we give a partial classification of submanifolds in $Q$, describe the degeneracy locus of its algebraic reduction and prove that the automorphism group of $Q$ satisfies the Jordan property.

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.

scholarly journals On the Classification of ALE Kähler Manifolds

2020 ◽  
Author(s):  
Hans-Joachim Hein ◽  
Rareş Răsdeaconu ◽  
Ioana Şuvaina
Keyword(s):  
Complex Structure ◽  
Kähler Manifold ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Quotient Singularity ◽  
Kahler Manifold ◽  
Kähler Surfaces

Abstract The underlying complex structure of an ALE Kähler manifold is exhibited as a resolution of a deformation of an isolated quotient singularity. As a consequence, there exist only finitely many diffeomorphism types of minimal ALE Kähler surfaces with a given group at infinity.

Uniqueness problem of meromorphic mappings of a complete Kähler manifold into a projective space

2020 ◽  
Vol 70 (4) ◽  
pp. 863-876
Author(s):  
Ha Huong Giang
Keyword(s):  
Projective Space ◽  
General Condition ◽  
General Position ◽  
Kähler Manifold ◽  
Uniqueness Problem ◽  
Uniqueness Theorems ◽  
Meromorphic Mappings ◽  
Kahler Manifold ◽  
Complete Kähler Manifold

AbstractIn this article, we prove a new generalization of uniqueness theorems for meromorphic mappings of a complete Kähler manifold M into ℙn(ℂ) sharing hyperplanes in general position with a general condition on the intersections of the inverse images of these hyperplanes.

Homogeneous locally conformally Kähler and Sasaki manifolds

2015 ◽  
Vol 26 (06) ◽  
pp. 1541001 ◽  
Author(s):  
D. V. Alekseevsky ◽  
V. Cortés ◽  
K. Hasegawa ◽  
Y. Kamishima
Keyword(s):  
Lie Groups ◽  
Reductive Group ◽  
Kähler Manifold ◽  
Symplectic Manifolds ◽  
Isotropy Group ◽  
Kahler Manifold ◽  
Locally Conformally Kähler ◽  
Reductive Lie Groups ◽  
Locally Conformally Symplectic ◽  
Sasaki Manifolds

We prove various classification results for homogeneous locally conformally symplectic manifolds. In particular, we show that a homogeneous locally conformally Kähler manifold of a reductive group is of Vaisman type if the normalizer of the isotropy group is compact. We also show that such a result does not hold in the case of non-compact normalizer and determine all left-invariant lcK structures on reductive Lie groups.

scholarly journals NO COHOMOLOGICALLY TRIVIAL NONTRIVIAL AUTOMORPHISM OF GENERALIZED KUMMER MANIFOLDS

2018 ◽  
Vol 239 ◽  
pp. 110-122
Author(s):  
KEIJI OGUISO
Keyword(s):  
Automorphism Group ◽  
Recent Work ◽  
Cohomology Group ◽  
Kähler Manifold ◽  
Nontrivial Automorphism ◽  
Kahler Manifold

For a hyper-Kähler manifold deformation equivalent to a generalized Kummer manifold, we prove that the action of the automorphism group on the total Betti cohomology group is faithful. This is a sort of generalization of a work of Beauville and a more recent work of Boissière, Nieper-Wisskirchen, and Sarti, concerning the action of the automorphism group of a generalized Kummer manifold on the second cohomology group.

Identities of the Riemannian Curvature Tensor of Almost C(ʎ)-Manifolds

2020 ◽  
pp. 49-54
Author(s):  
Aligadzhi R. Rustanov ◽  
Elena A. Polkina ◽  
Svetlana V. Kharitonova
Keyword(s):  
Curvature Tensor ◽  
Real Line ◽  
Kähler Manifold ◽  
Riemannian Curvature ◽  
Cosymplectic Manifold ◽  
The Real ◽  
Local Classification ◽  
Kahler Manifold ◽  
Riemannian Curvature Tensor

The geometry of the Riemannian curvature tensor of an almost C(λ)-manifold is studied. We have obtained several identities of the Riemannian curvature tensor of almost C(λ)-manifolds. Four additional identities are distinguished from these identities, on the basis of which four classes of almost C(λ)-manifolds are determined. A local classification of each of the distinguished classes of almost C(λ)-manifolds is obtained. It is proved that the set of almost C(λ)-manifolds of class R_1 coincides with the set of almost C(λ)-manifolds of class R_2, and it is also proved that the set of almost C(λ)-manifolds of class R_3 coincides with the set of almost C(λ)- manifolds of class R_4. We have found that an almost C(λ)-manifold, dimension greater than 3, is a manifold of class R_4 if and only if it is a cosymplectic manifold, i.e. when it is locally equivalent to the product of the Kähler manifold and the real line.

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