On compact holomorphically pseudosymmetric K

arXiv (Cornell University), 2023

Let (Ń , g, ∇) be a 2n-dimensional quasi-statistical manifold that admits a pseudo-Riemannian metric g (or h) and a linear connection ∇ with torsion. This paper aims to study an almost Hermitian structure (g, L) and an almost anti-Hermitian structure (h, L) on a quasi-statistical manifold that admit an almost complex structure L. Firstly, under certain conditions, we present the integrability of the almost complex structure L. We show that when d ∇ L = 0 and the condition of torsion-compatibility are satisfied, (Ń , g, ∇, L) turns into a Kähler manifold. Secondly, we give necessary and sufficient conditions under which (Ń , h, ∇, L) is an anti-Kähler manifold, where h is an anti-Hermitian metric. Moreover, we search the necessary conditions for (Ń , h, ∇, L) to be a quasi-Kähler-Norden manifold.

International Journal of Mathematics, 2012

Following T.-J. Li, W. Zhang , we continue to study the link between the cohomology of an almost-complex manifold and its almost-complex structure. In particular, we apply the same argument in and the results obtained by D. Sullivan in to study the cone of semi-Kähler structures on a compact semi-Kähler manifold.

Journal of Geometric Analysis, 2010

We study a special type of almost complex structures, called pure and full and introduced by T.J. Li and W. Zhang in [16], in relation to symplectic structures and Hard Lefschetz condition. We provide sufficient conditions to the existence of the above type of almost complex structures on compact quotients of Lie groups by discrete subgroups. We obtain families of pure and full almost complex structures on compact nilmanifolds and solvmanifolds. Some of these families are parametrized by real 2-forms which are anti-invariant with respect to the almost complex structures.

2009

In the paper we consider pseudo bihermitian structures - a pair of complex structures compatible with a pseudo Riemannian metric. As in the positive definite case we establish its relations with generalized (pseudo) Kaehler geometry and holomorphic Poisson structures. We provide a list of compact complex surfaces which could admit such structure and also examples of bihermitian structures on some of them. We also consider a naturally defined null plane distribution on generalized pseudo Kaehler 4-manifold and show that under a mild restriction it determines an Engel structure.

arXiv (Cornell University), 2022

Given a compact quantizable pseudo-Kähler manifold (M, ω) of constant signature, there exists a Hermitian line bundle (L, h) over M with curvature −2πi ω. We shall show that the asymptotic expansion of the Bergman kernels for L ⊗k-valued (0, q)-forms implies more or less immediately a number of analogues of well-known results, such as Kodaira embedding theorem and Tian's almost-isometry theorem.

Differential Geom Appl, 2000

An anti-Kaehlerian manifold is a complex manifold with an anti-Hermitian metric and a parallel almost complex structure. It is shown that a metric on such a manifold must be the real part of a holomorphic metric. It is proved that all odd Chern numbers of an anti-Kaehlerian manifold vanish and that complex parallelisable manifolds (in particular the factor space G/D of a complex Lie group G over the discrete subgroup D) are anti-Kaehlerian manifolds. A method of generating new solutions of Einstein equations by using the theory of anti-Kaehlerian manifolds is presented.

2016

We define the Kobayashi quotient of a complex variety by identifying points with vanishing Kobayashi pseudodistance between them and show that if a compact complex manifold has an automorphism whose order is infinite, then the fibers of this quotient map are nontrivial. We prove that the Kobayashi quotients associated to ergodic complex structures on a compact manifold are isomorphic. We also give a proof of Kobayashi's conjecture on the vanishing of the pseudodistance for hyperk\"ahler manifolds having Lagrangian fibrations without multiple fibers in codimension one. For a hyperbolic automorphism of a hyperk\"ahler manifold, we prove that its cohomology eigenvalues are determined by its Hodge numbers, compute its dynamical degree and show that its cohomological trace grows exponentially, giving estimates on the number of its periodic points.

Differential Geometry and its Applications, 2000

An anti-Kählerian manifold is a complex manifold with an anti-Hermitian metric and a parallel almost complex structure. It is shown that a metric on such a manifold must be the real part of a holomorphic metric. It is proved that all odd Chern numbers of an anti-Kählerian manifold vanish and that complex parallelisable manifolds (in particular the factor space G/D of a complex Lie group G over the discrete subgroup D ) are anti-Kählerian manifolds. A method of generating new solutions of Einstein equations by using the theory of anti-Kählerian manifolds is presented. *

1982

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The Journal of Geometric Analysis, 2020

Let p : X → Y be a surjective holomorphic mapping between Kähler manifolds. Let D be a bounded smooth domain in X such that every generic fiber D y := D ∩ p -1 (y) for y ∈ Y is a strongly pseudoconvex domain in X y := p -1 (y), which admits the complete Kähler-Einstein metric. This family of Kähler-Einstein metrics induces a smooth (1, 1)-form ρ on D. In this paper, we prove that ρ is positive-definite on D if D is strongly pseudoconvex. We also discuss the extensioin of ρ as a positive current across singular fibers.