On Real Structures of Rigid Surfaces

Abstract

We construct several rigid (i.e., unique in their deformation class) surfaces which have particular behavior with respect to real structures: in one example the surface has no any real structure, in the other one it has a unique real structure and this structure is not maximal with respect to the Smith-Thom inequality. So, it answers in negative to the following problems: existence of real surfaces in each complex deformation class and existence of maximal surfaces in each complex deformation class containing real surfaces. Besides, we prove that there is no real surfaces among the surfaces of general type with $p_g=q=0$ and $K^2=9$. As a by-product, the surfaces constructed give one more counterexample to "Dif=Def" problem.

Misprinted "REAL surfaces" in the title is replaced by "RIGID surfaces". Most of few improvements made are contained in Section 5

References in corpus (2)

• Topological properties of real algebraic varieties: du côté de chez Rokhlin
• Deformation inequivalent complex conjugated complex structures and applications

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