Ball and Surface Arithmetics by Rolf-Peter Holzapfel
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Ball and Surface Arithmetics

Ball and Surface Arithmetics


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About the Book

Bei höherdimensionalen komplexen Mannigfaltigkeiten stellt die Riemann-Roch-Theorie die grundlegende Verbindung von analytischen bzw. algebraischen zu topologischen Eigenschaften her. Dieses Buch befaßt sich mit Mannigfaltigkeiten der komplexen Dimension 2, d. h. mit komplexen Flächen. Hauptziel der Monographie ist es, neue rationale diskrete Invarianten (Höhen) in die Theorie komplexer Flächen explizit einzuführen und ihre Anwendbarkeit auf konkrete aktuelle Probleme darzustellen.Als erste unmittelbare Anwendung erhält man explizit und ganz allgemein Formeln vom Hurwitz-Typ endlicher Flächenüberlagerungen für die vier klassischen Invarianten, die auf andere Weise bisher nur in Spezialfällen zugänglich waren. Ein weiteres Anwendungsgebiet ist die Theorie der Picardschen Modulflächen: Neue Resultate werden beschrieben. Letztendlich kann im letzten Kapitel eine Ergänzung des bekannten Satzes von Bogomolov-Miyaoka-Yau mit Hilfe der Höhentheorie gezeigt werden.

The monograph presents basically an arithmetic theory of orbital surfaces with cusp singularities. As main invariants orbital hights are introduced, not only for the surfaces but also for the components of orbital cycles. These invariants are rational numbers with nice functorial properties allowing precise formulas of Hurwitz type and a fine intersection theory for orbital cycles. For ball quotient surfaces they appear as volumes of fundamental domains. In the special case of Picard
modular surfaces they are discovered by special value of Dirichlet L-series or higher Bernoulli numbers. As a central point of the monograph a general Proportionality Theorem in terms of orbital hights is proved. It yields a strong criterion to decide effectively whether a surface with given cycle supports a ball quotient structure being Kaehler-Einstein with negative constant holomorphic sectional curvature outside of this cycle. The theory is applied to the classification of Picard modular surfaces and to surfaces geography.
About the Author: Dr. Rolf-Peter Holzapfel ist Mitglied der Arbeitsgruppe Allgebraische Geometrie und Zahlentheorie am Max-Planck-Institut zur Förderung der Wissenschaften, Berlin.


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Product Details
  • ISBN-13: 9783322901712
  • Publisher: Vieweg+teubner Verlag
  • Publisher Imprint: Vieweg+teubner Verlag
  • Height: 229 mm
  • No of Pages: 414
  • Series Title: Aspects of Mathematics
  • Weight: 625 gr
  • ISBN-10: 3322901718
  • Publisher Date: 19 May 2012
  • Binding: Paperback
  • Language: English
  • Returnable: N
  • Spine Width: 22 mm
  • Width: 152 mm


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