Generalized Cartan-Behnke-Stein's theorem and $q$-pseudoconvexity in a Stein manifold

2020 ◽  
Vol 72 (4) ◽  
pp. 527-535
Author(s):  
Shun Sugiyama
Keyword(s):  
Stein Manifold

Stein manifolds of nonnegative curvature

2018 ◽  
Vol 18 (3) ◽  
pp. 285-287
Author(s):  
Xiaoyang Chen
Keyword(s):  
Fixed Point ◽  
Sectional Curvature ◽  
Normal Bundle ◽  
Stein Manifold ◽  
Riemannian Metric ◽  
Nonnegative Curvature ◽  
Fixed Point Set ◽  
Nonnegative Sectional Curvature ◽  
Point Set ◽  
Nonempty Compact

AbstractLet X bea Stein manifold with an anti-holomorphic involution τ and nonempty compact fixed point set Xτ. We show that X is diffeomorphic to the normal bundle of Xτ provided that X admits a complete Riemannian metric g of nonnegative sectional curvature such that τ*g = g.

scholarly journals Holomorphic Vector Bundles on Holomorphically Convex Complex Manifolds

2006 ◽  
Vol 13 (1) ◽  
pp. 7-10
Author(s):  
Edoardo Ballico
Keyword(s):  
Vector Bundle ◽  
Complex Manifold ◽  
Vector Bundles ◽  
Open Neighborhood ◽  
Holomorphic Vector Bundle ◽  
Stein Manifold ◽  
Complex Manifolds ◽  
Holomorphic Vector Bundles

Abstract Let 𝑋 be a holomorphically convex complex manifold and Exc(𝑋) ⊆ 𝑋 the union of all positive dimensional compact analytic subsets of 𝑋. We assume that Exc(𝑋) ≠ 𝑋 and 𝑋 is not a Stein manifold. Here we prove the existence of a holomorphic vector bundle 𝐸 on 𝑋 such that is not holomorphically trivial for every open neighborhood 𝑈 of Exc(𝑋) and every integer 𝑚 ≥ 0. Furthermore, we study the existence of holomorphic vector bundles on such a neighborhood 𝑈, which are not extendable across a 2-concave point of ∂(𝑈).

HOLOMORPHIC FILLING OF ℝP3

2000 ◽  
Vol 02 (03) ◽  
pp. 349-363 ◽  
Author(s):  
RICHARD HIND
Keyword(s):  
Stein Manifold ◽  
Pseudoconvex Boundary

We will demonstrate that up to Stein homotopy there is a unique Stein manifold which has a strictly pseudoconvex boundary diffeomorphic to ℝP3. The manifold is a complexification of S2.

scholarly journals Chern Classes of Projective Modules

1963 ◽  
Vol 23 ◽  
pp. 121-152 ◽  
Author(s):  
Hideki Ozeki
Keyword(s):  
Vector Bundle ◽  
Cross Sections ◽  
Chern Class ◽  
Vector Bundles ◽  
Stein Manifold ◽  
Projective Modules ◽  
Singular Variety ◽  
Differentiable Functions ◽  
Holomorphic Vector Bundles ◽  
Differentiable Vector

In topology, one can define in several ways the Chern class of a vector bundle over a certain topological space (Chern [2], Hirzebruch [7], Milnor [9], Steenrod [15]). In algebraic geometry, Grothendieck has defined the Chern class of a vector bundle over a non-singular variety. Furthermore, in the case of differentiable vector bundles, one knows that the set of differentiable cross-sections to a bundle forms a finitely generated projective module over the ring of differentiable functions on the base manifold. This gives a one to one correspondence between the set of vector bundles and the set of f.g.-projective modules (Milnor [10]). Applying Grauert’s theorems (Grauert [5]), one can prove that the same statement holds for holomorphic vector bundles over a Stein manifold.

Remarks on k-Leviflat Complex Manifolds

1979 ◽  
Vol 31 (4) ◽  
pp. 881-889 ◽  
Author(s):  
B. Gilligan ◽  
A. Huckleberry
Keyword(s):  
Plurisubharmonic Function ◽  
Stein Manifold ◽  
Levi Form ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Complex Manifolds ◽  
Generic Fiber ◽  
Convex Case ◽  
Compact Complex Manifolds

In the theory of functions of several complex variables one is naturally led to study non-compact complex manifolds which have certain types of exhaustions. For example, on a Stein manifold X there is a strictly plurisubharmonic function ϕ: X → R+ so that the pseudoballs Bc = {φ < c } exhaust X. Conversely, a manifold which has such an exhaustion is Stein. The purpose of this note is to study a class of manifolds which have exhaustions along the lines of those on holomorphically convex manifolds, namely the k-Leviflat complex manifolds. Unlike the Stein case, the Levi form may have positive dimensional 0-eigenspaces. In the holomorphically convex case these are tangent to the generic fiber of the Remmert reduction.

COMPLEXIFICATIONS OF HOLOMORPHIC ACTIONS AND THE BERGMAN METRIC

2004 ◽  
Vol 15 (08) ◽  
pp. 735-747 ◽  
Author(s):  
ANDREA IANNUZZI ◽  
ANDREA SPIRO ◽  
STEFANO TRAPANI
Keyword(s):  
Lie Groups ◽  
Complex Manifold ◽  
Analogous Result ◽  
Stein Manifold ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Bergman Metric ◽  
Stein Manifolds ◽  
Invariant Domain ◽  
Necessary And Sufficient

Let G=(ℝ,+) act by biholomorphisms on a Stein manifold X which admits the Bergman metric. We show that X can be regarded as a G-invariant domain in a "universal" complex manifold X* on which the complexification [Formula: see text] of G acts. The analogous result holds for actions of a larger class of real Lie groups containing, e.g. abelian and certain nilpotent ones. For holomorphic actions of such groups on Stein manifolds, necessary and sufficient conditions for the existence of X* are given.

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