On Determining Sets in a Stein Manifold

1965 ◽  
pp. 48-58 ◽  
Author(s):  
A. Aeppli
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.

STRATIFIED TRANSVERSALITY OF HOLOMORPHIC MAPS

2013 ◽  
Vol 24 (13) ◽  
pp. 1350106 ◽  
Author(s):  
SAURABH TRIVEDI
Keyword(s):  
Weak Topology ◽  
Stein Manifold ◽  
Real Case ◽  
Holomorphic Maps ◽  
Countable Collection ◽  
Stein Manifolds ◽  
The Real ◽  
Oka Manifold ◽  
The Stability ◽  
Necessary And Sufficient

We discuss genericity and stability of transversality of holomorphic maps to complex analytic stratifications. We prove that the set of maps between Stein manifolds and Oka manifolds transverse to a countable collection of submanifolds in the target is dense in the space of holomorphic maps with the weak topology. This greatly generalizes earlier results on the genericity of transverse maps by Forstnerič and by Kaliman and Zaidenberg. As an application we show that the Whitney (a)-regularity of a complex analytic stratification is necessary and sufficient for the stability of transverse holomorphic maps between a Stein manifold and an Oka manifold. This gives an analogue of a theorem in the real case due to Trotman.

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