A. Degtyarev, I. Itenberg, V. Kharlamov
Real Enriques Surfaces
Lecture Notes in Mathematics, 1746
SpringerVerlag, 2000
To Vicky, To Natasha, To Sonia 

The book is the first attempt of a systematic study of real Enriques
surfaces, which culminates in their classification
up to deformation. Simple explicit topological invariants are elaborated
for identifying the deformation classes of real Enriques surfaces.
Some of the invariants are new and can be applied to other classes
of surfaces or higherdimensional varieties.
The book, which is intended to researches and graduate students in
real algebraic geometry, can as well be of interest to all those who want
to acquire a familiarity with real algebraic geometry and its techniques.
The study relies on topology of involutions, arithmetics of integral
quadratic forms, algebraic geometry of surfaces, and the hyperkähler
structure on K3surfaces. A comprehensive summary of the necessary results
and techniques from each of these fields is included in the book. Some
of the results are developed further. Among them worth mentioning are
a detailed study of integral lattices with a pair of commuting
involutions and that of a certain class of rational complex surfaces.
Mathematical Subject Classification:
14P25, 14J28, 14J15, 14J50, 14J80, 57S17, 58D27
Keywords and Phrases:
Enriques surfaces,
real algebraic surfaces,
deformation of surfaces,
hyperkähler structure,
topology of real algebraic varieties
Level and Target Group:
Monograph (with elements of survey);
Graduate students and researchers
Table of Contents
Introduction
Part I. Tools
Chapter I. Topology of involutions
1. Equivariant homology
2. Involutions on manifolds
Chapter II. Integral lattices and quadratic forms
3. Finite quadratic forms
4. Gluing forms
5. Integral lattices
Chapter III. Algebraic surfaces
6. Common facts and notation
7. DPNsurfaces and pairs
Chapter IV. Real surfaces: the topological aspects
8. Principal prohibitions
Part II. Enriques surfaces
Summary: Deformation classes
Chapter V. Topology of real Enriques surfaces
9. Homology of a generalized real Enriques surface
10. Gluing eigenlattices
11. Topological classification
12. Calculation of the PontrjaginViro form
Chapter VI. Moduli of real Enriques surfaces
13. Periods of $K3$surfaces
14. Periods of real Enriques surfaces
15. Quaternionic structure and Donaldson's trick
16. The fundamental polyhedron and topology of the real point set
Chapter VII. Deformation types: the hyperbolic and parabolic cases
17. Del Pezzo surfaces
18. Almost Del Pezzo surfaces ((g,r)surfaces, g \ge 3, r \ge 1)
19. Surfaces with a genus 2 component ((2,r)surfaces, r \ge 1)
20. Surfaces with a genus 1 component ((1,r)surfaces)
Chapter VIII. Deformation types: the elliptic and parabolic cases
21. Invariants and preliminary results
22. Classification of actions
Appendix A. Beginner's manual on the moduli of real curves and surfaces
A1. Hyperelliptic curves on rational ruled surfaces
A2. Rigid isotopies of curves on surfaces
A3. Surfaces
Appendix B. Horikawa models
Appendix C. Determination of real Enriques surfaces
Appendix D. Finiteness Results
Bibliography
Glossary
Index
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