Real Enriques Surfaces

by Degtyarev, A.; Itenberg, I.; Kharlamov, V.

• ISBN13: 9783540410881
• ISBN10: 3540410880
• eBook ISBN(s): 9783540399483
• Format: Hardcover
• Copyright: 2000-10-01
• Publisher: Springer Verlag

Summary

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 of theses are new and can be applied to other classes of surfaces or higher-dimensional varieties. Intended for researchers and graduate students in real algebraic geometry it may also interest others who want to become familiar with the field and its techniques. The study relies on topology of involutions, arithmetics of integral quadratic forms, algebraic geometry of surfaces, and the hyperkähler structure of K3-surfaces. A comprehensive summary of the necessary results and techniques from each of these fields is included. Some results are developed further, e.g., a detailed study of lattices with a pair of commuting involutions and a certain class of rational complex surfaces.

Table of Contents

Introduction	vii
Part I Tools	1	(87)
Topology of involutions	2	(27)
Equivariant homology	2	(16)
Involutions on manifolds	18	(11)
Integral lattices and quadratic forms	29	(24)
Finite quadratic forms	29	(11)
Gluing forms	40	(6)
Integral lattices	46	(7)
Algebraic surfaces	53	(26)
Common facts and notation	54	(12)
DPN-surfaces and pairs	66	(13)
Real surfaces: the topological aspects	79	(9)
Principal prohibitions	79	(9)
Part II. Enriques Surfaces	88	(103)
Summary: Deformation classes	89	(8)
Topology of real Enriques surfaces	97	(30)
Homology of a generalized real Enriques surface	98	(11)
Gluing eigenlattices	109	(6)
Topological classification	115	(4)
Calculation of the Pontrjagin-Viro form	119	(8)
Moduli of real Enriques surfaces	127	(18)
Periods of K 3-surfaces	127	(7)
Periods of real Enriques surfaces	134	(3)
Quaternionic structure and Donaldson's trick	137	(3)
The fundamental polyhedron and topology of the real point set	140	(5)
Deformation types: the hyperbolic and parabolic cases	145	(24)
Del Pezzo surfaces	146	(3)
Almost Del Pezzo surfaces ((g, r)-surfaces, g ≥ 3, r ≥ 1)	149	(6)
Surfaces with a genus 2 component ((2, r)-surfaces, r ≥ 1)	155	(3)
Surfaces with a genus 1 component ((1, r)-surfaces)	158	(11)
Deformation types: the elliptic and parabolic cases	169	(22)
Invariants and preliminary results	170	(9)
Classification of actions	179	(12)
Appendix A. Beginner's manual on the moduli of real curves and surfaces	191	(22)
A1. Abstract curves	191	(1)
A2. Hyperelliptic curves on rational ruled surfaces	192	(9)
A3. Rigid isotopies of curves on surfaces	201	(10)
A4. Surfaces	211	(2)
Appendix B. Horikawa models	213	(9)
Appendix C. Determination of real Enriques surfaces	222	(10)
Appendix D. Finiteness results	232	(10)
Bibliography	242	(7)
Glossary	249	(5)
Index	254