Holomorphic curves in complex spaces

By Barbara Drinovec-Drnovšek, F. Forstnerič. 2006

78 citations recorded by the source corpus.

We study the existence of topologically closed complex curves normalized by bordered Riemann surfaces in complex spaces. Our main result is that such curves abound in any noncompact complex space admitting an exhaustion function whose Levi form has at least two positive eigenvalues at every point outside a compact set, and this condition is essential. The proof involves a lifting method for the boundary of the curve and a newly developed technique of gluing holomorphic sprays over Cartan pairs in Stein manifolds whose value lie in a complex space, with control up to the boundary of the domains. (The latter technique is also exploited in the subsequent papers math.CV/0607185 and math.CV/0609706.) We also prove that any compact complex curve with C^2 boundary in a complex space admits a basis of open Stein neighborhoods. In particular, an embedded disc of class C^2 with holomorphic interior in a complex manifold admits a basis of open polydisc neighborhoods.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c75bcc9b36c9464d61bd04634073c02ffe11a5c4

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In Applied Mathematics

1. E. F. Wold
2. J. Fornaess
3. E. Porten
4. Joël Merker
5. F. Forstnerič
6. F. Lárusson
7. Y. Siu
8. F. Kutzschebauch

In Geometry and Topology

1. E. F. Wold
2. J. Fornaess
3. E. Porten
4. Joël Merker
5. F. Forstnerič
6. F. Lárusson
7. Y. Siu
8. Jasna Prezelj

Citations in the corpus

• Holomorphic Approximation: The Legacy of Weierstrass, Runge, Oka–Weil, and Mergelyan (2018, background): “ions of class Cr (r ∈N)on compact or closed sets in a Riemannsurface. Such problems arise not only in complexanalysis (for instance, in constructionsof closed complex curves in complex manifolds, see [44], or in constructions of proper holomorphic embeddings of open Riemann surfaces into C2, see [66”
• Manifolds of holomorphic mappings from strongly pseudoconvex domains (2006, methodology): “e interior of the respective set, endowed with the sup-norm. By [10, Lemma 3.4] there exist continuous linear operators A: Cr′ → Ar′, B: Cr′ → Br′ satisfying (3.2) c= A(c) −B(c), c∈ Cr′. The proof in [10] uses a linear solution operator for the ∂¯-equation on the level of (0,1)-forms on Dsatisfying ”
• The Hartogs extension theorem on (n – 1)-complete complex spaces (2007, methodology): “ntirely ins ide Xreg. This is possible thanks to an idea of Demailly which consists of modifying the initial exhaustion ρ so that Xsing is put at −∞. A recent application of this idea also appears in [4]. THE HARTOGS EXTENSION THEOREM ON (n − 1)-COMPLETE COMPLEX SPACES 7 3.3. Putting Xsing into a we”