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Real Enriques surfaces. (English) Zbl 0963.14033
Lecture Notes in Mathematics. 1746. Berlin: Springer. xvi, 259 p. (2000).
The subject of this remarkable book is a particular case of the second part of Hilbert’s 16th problem understood as the problem of classification of algebraic or analytic varieties with real structures (anti-holomorphic involutions) up to equivariant topological, isotopy or deformation equivalence. Real Enriques surfaces represent one of very few classes of varieties which can be classified completely (other examples are plane curves of small degrees, rational, abelian or K3 surfaces). The contents of the monograph is of interest from various points of view. Concrete classification results reflect important geometric phenomena. One of them is that the deformation class of a real Enriques surface is determined by the topology of its complex conjugation involution. Similar result holds for other classified classes of real algebraic and analytic surfaces. – Real Enriques surfaces of some types possess specific geometric structures, for example, real Enriques surfaces without real points, or their quotients by the complex conjugation are 4-dimensional Einstein manifolds extremal in the sense of the Hitchin inequality $$|\sigma(E)|\leq\frac{2}{3}\chi(E)$$.
Methods for the study of real Enriques surfaces are set forth in a systematic and self-contained form which allows the reader to enjoy an interplay of algebraic topology, algebraic geometry, arithmetic of quadratic forms, some specific tools of real algebraic geometry. Among them one finds the topology of involutions, including related spectral sequences and dualities, finite 4-periodic quadratic forms, algebraic geometry of K3 and Enriques surfaces. In fact, the techniques developed apply to other classes of real algebraic varieties.
The authors discuss possible applications, related topics and open questions. In particular, they pose two “finiteness” questions: Is it true that a given complex variety admits at most finitely many real structures (up to conjugation by automorphisms of the variety)? Is it true that a given complex analytic variety admits at most finitely many holomorphic involutions (up to conjugation by automorphisms of the variety)?

##### MSC:
 14P25 Topology of real algebraic varieties 14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry 14J28 $$K3$$ surfaces and Enriques surfaces 14J80 Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants) 57S17 Finite transformation groups 58D27 Moduli problems for differential geometric structures
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