On Real Structures of Rigid Surfaces

Details On Real Structures of Rigid Surfaces On Real Structures of Rigid Surfaces

2001-01-11

arxiv.org/abs/math/0101098v2

Mathematics

Algebraic Geometry

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

Scientific paper

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.

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