Real Enriques Surfaces
This is the first attempt of a systematic study of real Enriques surfaces culminating in their classification up to deformation. Simple explicit topological invariants are elaborated for identifying the deformation classes of real Enriques surfaces. Some
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1746
Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo
A. Degtyarev 1. Itenberg V. Kharlamov
Real Enriques Surfaces
Springer
Author Alexander Degtyarev Bilkent University 06533 Ankara, Turkey E-mail: [email protected] Ilia Itenberg Institut de Recherche Mathernatique de Universite de Rennes (CNRS) 35042 Rennes Cedex, France E-mail: [email protected] Viatcheslav Kharlamov Universite Louis Pasteur et IRMA (CNRS) 7 rue Rene Decartes 67084 Strasbourg Cedex, France E-mail: [email protected] The cover ilustration is taken from http://enriques.mathematik.uni.mainz.delkonldocs/enriques.shtml with kind permission of the authors W. Barth and S. EndraB
Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Degtyarev, Aleksandr: Real Enriques surfaces I A. Degtyarev ; I. Itenberg ; V. Kharlamov. Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris ; Singapore; Tokyo: Springer, 2000 (Lecture notes in mathematics; 1746) ISBN 3-540-41088-0
Mathematics Subject Classification (2000): 14P25, 14128, 14J15, 14J80, 57S 17.58027 ISSN 0075-8434 ISBN 3-540-41088-0 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer- Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science+Business Media GmbH © Springer-Verlag Berlin Heidelberg 2000 Printed in Germany Typesetting: Camera-ready TEX output by the author SPIN: 10724240 41/3142-543210 - Printed on acid-free paper
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Introduction
Enriques surfaces playa special role in the theory of surfaces, both algebraic and analytic. They form a separate class in the Enriques-Kodaira classification of minimal surfaces: it is one of the four classes of Kodaira dimension 0 (the three others are abelian, hyperelliptic, and K3-surfaces). For an algebraist, a particular feature of Enriques surfaces is that they are irrational and have no holomorphic differential forms. For a topologist, they are the simplest examples of smooth 4-manifolds with even intersection form and signature not divisible by 16. The fundamental group of an Enriques surface is Z2. Its universal covering is a K3-surface and, vice versa, the orbit space of a fixed point free holomorphic involution on a K3-surface is an Enriques surface. To visualize an Enriques surface one can pick a nonsingular curve D C pI X pI of bidegree (4, 4) invariant with respect to 8 X 8, where 8: pI -t pI
