D* extension property without hyperbolicity

Manuscripta Mathematica, 2013

Using the concept of inner integral curves defined by Hirschowitz we generalize a recent result by Kim, Levenberg and Yamaguchi concerning the obstruction of a pseudoconvex domain spread over a complex homogeneous manifold to be Stein. This is then applied to study the holomorphic reduction of pseudoconvex complex homogeneous manifolds X = G/H. Under the assumption that G is solvable or reductive we prove that X is the total space of a G-equivariant holomorphic fiber bundle over a Stein manifold such that all holomorphic functions on the fiber are constant.

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.

Siberian Mathematical Journal, 1975

Any p r o p e r holomorphic self-map of polydisk U n ~ C n is rational [1]. This mapping, of course, may not be in a one-to-one manner, i. e. , it is biholomorphic. Eisenman in [2] proved that proper rational selfmapping of the ball B n ~ C n (n > 1) is biholomorphic. Such a difference of a ball from the polydisk is apparently explained by a strict pseudoconvexity of the bali. In this work we shall emphasize the results of Eisenman by proving that if D 1 and D 2 are strictly pseud0convex domains in C n, then proper holomorphie mapping 1: D~-*.Dz, (1) which is extended to mapping f : I) i-* I~ 2 of class C', is locally biholomorphic. If apart from this, D i = D2, the mapping f is biholomorphic. We shall note that at D~ # D 2, mapping (1), generally speaking, may not 9 be globally biholomorphic. F o r example, domains D 1 = {z ~ C 2 : Iz 112 + Iz 214 + Iz z t-4 > 3} and D 2 = {z 6 C 2 : Iz 112 + Iz2l 2 + Iz2 I-2 < 3} are strictly pseudoconvex, and the proper holomorphic mapping f = (fl, f2): D1 ~ D2, fl(z) = zl, f2(z) = z~ is not biholomorphic.

Transactions of the American Mathematical Society, 1977

This paper deals with regularity properties of the infinitesimal form of the Kobayashi pseudo-distance. This form is shown to be upper semicontinuous in the parameters of a deformation of a complex manifold. The method of proof involves the use of a parametrized version of the Newlander-Nirenberg Theorem together with a theorem of Royden on extending regular mappings from polydiscs into complex manifolds. Various consequences and improvements of this result are discussed; for example, if the manifold is compact hyperbolic the infinitesimal Kobayashi metric is continuous on the union of the holomorphic tangent bundles of the fibers of the deformation. This result leads to the fact that the coarse moduli space of a compact hyperbolic manifold is Hausdorff. Finally, the infinitesimal form is studied for a class of algebraic manifolds which contains algebraic manifolds of general type. It is shown that the form is continuous on the tangent bundle of a manifold in this class. Many member...

2009

For compact Kählerian manifolds, the holomorphic pseudosymmetry reduces to the local symmetry if additionally the scalar curvature is constant and the structure function is non-negative. Similarly, the holomorphic Ricci-pseudosymmetry reduces to the Ricci-symmetry under these additional assumptions. We construct examples of non-compact essentially holomorphically pseudosymmetric Kählerian manifolds. These examples show that the compactness assumption cannot be omitted in the above stated theorem. Recently, the first examples of compact, simply connected essentially holomorphically pseudosymmetric Kählerian manifolds are discovered in [4]. In these examples, the structure functions change their signs on the manifold.

Mathematische Annalen, 1988

Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 2017

Our aim in this paper is to characterize smooth domains (D, J) and (D , J) in almost complex manifolds of real dimension 2n + 2 with a covering orbit { f k (p)}, accumulating at a strongly pseudoconvex boundary point, for some (J, J)-holomorphic coverings f k : (D, J) → (D , J) and p ∈ D. It was shown that such domains are both biholomorphic to a model domain, if the source domain (D, J) admits a bounded strongly Jplurisubharmonic exhaustion function. Furthermore, if the target domain (D , J) is strongly pseudoconvex, then both (D, J) and (D , J) are biholomorphic to the unit ball in C n+1 with the standard complex structure. Our results can be considered as compactness theorems for sequences of pseudo-holomorphic coverings.

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.

Indiana University Mathematics Journal, 2009

Pseudoconvexity of a domain in C n is described in terms of the existence of a locally defined plurisubharmonic/holomorphic function near any boundary point that is unbounded at the point.

Mathematische Annalen, 1982

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.

We prove that the core of a complex manifold X is the union of pairwise disjoint pseudoconcave sets on which all uniformly bounded continuous plurisubharmonic functions on X are constant. Similarly, the minimal kernel of a weakly complete complex manifold decomposes into the union of compact pseudoconcave sets on which all continuous plurisub-harmonic functions are constant. Versions of these results for standard smoothness classes are obtained. Analogous facts are discussed in the context of Richberg's regularization of continuous strongly plurisubharmonic functions.

2000

Our goal here is to prove that each point in an almost-complex surface has a basis of complete hyperbolic neighborhoods. The problem is local, and therefore we can consider the case when our surface is R 4 with an arbitrary almost-complex structure J. Throughout this paper, almost-complex structures are assumed to be of class C 1,α for some 0 < α < 1. Let C be some non-singular J-complex curve passing through the origin.

Proceedings of the American Mathematical Society, 1994

The purpose of this paper is to study the hyperbolicity and the tautness of spaces that have the Schottky property. Moreover, the hyperbolicity of compact complex spaces is characterized by the classical theorem of Bloch.

Annali di Matematica Pura ed Applicata, 1989

Harris and ~ei]]en on Kobayashi and Carathdodory metrics on complex mani]olds are obtained. In particular we prove that on every co~lex mani]old (]inite or in]inite.dimensional) the Kobayashi distance is the integrated /orm o/ the corresponding infinitesimal metric.