Real Algebraic Surfaces

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1392

Robert Silhol

Real Algebraic Surfaces

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

1392

Robert Silhol

Real Algebraic Surfaces

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong

Author Robert Silhol Institut de Mathematiques, Universite des Sciences et Techniques du Languedoc 34060 Montpellier Cedex, France

Mathematics Subject Classification (1980): 14G30, 14J26, 14J27, 14J28, 14K05, 14K 10 ISBN 3-540-51563-1 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-51563-1 Springer-Verlag New York Berlin Heidelberg

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©! Springer-Verlag Berlin Heidelberg 1989 Printed in Germany Printing and binding: Druckhaus Beltz. Hemsbach/Bergstr. 2146/3140-543210 - Printed on acid-free paper

INTRODUCTION

These

notes are centred on one question : given a real algebraic

surface X determine the topology of the real part X(R). Of

course, since,

algebraic real

geometry is

to quote

the classification

algebraic geometry

notes.

Hartshorne, the

also), the

guiding problem in

problem (and this goes for

latter is

very present in these

In fact it is present to the point, that we have only obtained

a precise answer to our original question when we have obtained a precise answer to the classification problem. In this sense one could say that the underlying theme (and even, the main theme) of these notes is the classification problem of real algebraic surfaces. This

second preoccupation

has dictated

the plan of these notes

and to some extent the methods used. We explain this. If two algebraic varieties are real isomorphic, then they certainly are complex isomorphic. Hence, our starting point, the well known Enriques­Kodaira classification of complex algebraic surfaces, and the plan. To

be able to make the most of the knowledge accumulated on com-

plex

algebraic surfaces we

have used

an alternative definition for

real

algebraic varieties, explicitly, we define them as complex alge-

braic varieties with an antiholomorphic involution. Otherwise said, we consider an

real algebraic varieties as complex algebraic varieties with

action of

only

the Galois

preoccupation, the

group Gal(CIR)

(in the projective case, our

two definitions

are equivalent ­ see I.§l).

This is the foundation of all the methods used in these notes. From classes,

this point those

H*(X(R),Z/2)

for

of view

real algebraic

surfaces fall into two

which t