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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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
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