Proper holomorphic disks in the complement of varieties in $\mathbb{C}^2$

Borell, Stefan; Kutzschebauch, Frank; Wold, Erlend Fornaess

doi:10.4310/mrl.2008.v15.n4.a18

Cited by 2 publications

(2 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…there also exist embedded holomorphic discs with this property according to Borell et al [8], and for dim Y ≥ 3, this holds by the general position argument. Proper holomorphic discs in C 2 with images contained in certain concave cones were constructed by Globevnik and the second named author [23] in 2001.…”

Section: Introductionmentioning

confidence: 65%

Proper Holomorphic Maps in Euclidean Spaces Avoiding Unbounded Convex Sets

Drnovsek

Forstnerič

2023

J Geom Anal

View full text Add to dashboard Cite

We show that if E is a closed convex set in $${\mathbb {C}}^n$$ C n $$(n>1)$$ ( n > 1 ) contained in a closed halfspace H such that $$E\cap bH$$ E ∩ b H is nonempty and bounded, then the concave domain $$\Omega = {\mathbb {C}}^n{\setminus } E$$ Ω = C n \ E contains images of proper holomorphic maps $$f:X\rightarrow {\mathbb {C}}^n$$ f : X → C n from any Stein manifold X of dimension $$<n$$ < n , with approximation of a given map on closed compact subsets of X. If in addition $$2\dim X+1\le n$$ 2 dim X + 1 ≤ n then f can be chosen an embedding, and if $$2\dim X=n$$ 2 dim X = n , then it can be chosen an immersion. Under a stronger condition on E, we also obtain the interpolation property for such maps on closed complex subvarieties.

show abstract

Section: Introductionmentioning

confidence: 65%

Proper Holomorphic Maps in Euclidean Spaces Avoiding Unbounded Convex Sets

Drnovsek

Forstnerič

2023

J Geom Anal

View full text Add to dashboard Cite

show abstract

“…More generally, one can ask which type of sets in Stein manifolds can be avoided by proper holomorphic maps from Stein manifolds of sufficiently low dimension. In this direction, Drinovec Drnovšek showed in [37] that any closed complete pluripolar set can be avoided by proper holomorphic discs; see also Borell et al [24] for embedded discs in C n . Note that every closed complex subvariety is a complete pluripolar set.…”

Section: Embeddings Into Stein Manifolds With the Density Propertymentioning

confidence: 99%

Holomorphic Embeddings and Immersions of Stein Manifolds: A Survey

Forstnerič

2018

Springer Proceedings in Mathematics &Amp; Statistics

View full text Add to dashboard Cite

In this paper we survey results on the existence of holomorphic embeddings and immersions of Stein manifolds into complex manifolds. Most of them pertain to proper maps into Stein manifolds. We include a new result saying that every continuous map X → Y between Stein manifolds is homotopic to a proper holomorphic embedding provided that dim Y > 2 dim X and we allow a homotopic deformation of the Stein structure on X.

show abstract

Proper holomorphic disks in the complement of varieties in $\mathbb{C}^2$

Abstract: For any analytic set X ⊂ C 2 there exists a proper holomorphic embedding ϕ : △ ֒→ C 2 such that ϕ(△) ∩ X = ∅.

Cited by 2 publications

References 7 publications

Proper Holomorphic Maps in Euclidean Spaces Avoiding Unbounded Convex Sets

Proper Holomorphic Maps in Euclidean Spaces Avoiding Unbounded Convex Sets

Holomorphic Embeddings and Immersions of Stein Manifolds: A Survey

Contact Info

Product

Resources

About