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Surfaces with \(\mathrm{DIF}\neq \mathrm{DEF}\) real structures. (English. Russian original) Zbl 1153.14027

Izv. Math. 70, No. 4, 769-807 (2006); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 70, No. 4, 135-174 (2006).
A real structure on a complex manifold \(X\) is an anti-holomorphic automorphism \(\sigma:X \to X\) such that \(\sigma^{2}=Id_{X}\). A Kodaira-Spencer deformation of complex surfaces \(p : Z \to D\) is said a real deformation if \(Z\) has a real structure, \(D \subset \mathbb{C}\) is a disk and \(p\) is equivariant. A complex surface equipped with a real structure is called a real surface. Two real surfaces \(X'\) and \(X''\) are called deformation equivalent if they can be connected by a chain \(X'=X_0\), …, \(X_k=X''\) so that \(X_i\) and \(X_{i-1}\) are isomorphic to real fibers of a real deformation. A real surface \(X\) is called quasi-simple if it is deformation equivalent to any other real surface \(X'\) such that \(X'\) is deformation equivalent to \(X\) as a complex surface, and the real structure of \(X'\) is diffeomorphic to that of \(X\). In this paper the authors study real Campedelli surfaces up to real deformations and give several examples of such surfaces which are not quasi-simple, i.e. the real DIF=DEF problem has a negative answer for surfaces of general type. These are the first examples of real surfaces which are not quasi-simple, since for all types of real surfaces until now classified the real DIF=DEF problem has a positive answer. However, note that there are examples of surfaces of general type which are quasi-simple: the Miyaoka-Yau surfaces, that is, surfaces covered by a ball in \(\mathbb{C}^{2}\); they satisfy the relation \(c^{2}_{1}=3c_{2}\).

MSC:

14J15 Moduli, classification: analytic theory; relations with modular forms
14J29 Surfaces of general type
14N20 Configurations and arrangements of linear subspaces
32G05 Deformations of complex structures
14P25 Topology of real algebraic varieties
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