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On the hyperbolicity of the complements of curves in algebraic surfaces: The three-component case. (English) Zbl 0847.32028
Die Arbeit erzielt einen beachtlichen Fortschritt zu der Frage, wann zweidimensionale komplexe Mannigfaltigkeiten eine hyperbolische Struktur besitzen. Die Anwort fällt positiv aus für die projektive Ebene, aus der man eine “generische” Konfiguration dreier Kurven vom Grad \(\geq 2\), eine davon vom Grad \(\geq 3\), herausnimmt. Was “generisch” heißt, läßt sich weiter präzisieren. Ein analoges Resultat erhalten die Autoren für generische glatte Hyperflächen des projektiven Raums, aus denen man drei Kurven genügend hohen Grades herausnimmt.
Beweistechnisch beruhen die Resultate teilweise auf Rechnungen mit Invarianten komplexer Mannigfaltigkeiten, teilweise auf Sätzen über Werteverteilung: Es ist nur zu zeigen, daß keine nichtkonstante holomorphe Abbildung vom Grad \(\leq 2\) in die fragliche Mannigfaltigkeit existiert.

MSC:
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
32H30 Value distribution theory in higher dimensions
14J25 Special surfaces
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
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