A. Degtyarev, I. Itenberg, V. Kharlamov

Real Enriques Surfaces

Lecture Notes in Mathematics, 1746
Springer-Verlag, 2000



To Vicky, To Natasha, To Sonia    W.Barth, S.Endraß

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 higher-dimensional 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 K3-surfaces. 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. DPN-surfaces 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 Pontrjagin-Viro 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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