On Real Structures of Rigid Surfaces

TL;DR

This paper constructs and analyzes rigid complex surfaces with specific real structure properties, providing counterexamples to existing conjectures and exploring the existence and maximality of real structures in complex deformation classes.

Contribution

It introduces new examples of rigid surfaces with unique or no real structures, addressing open problems about their existence and maximality in deformation classes.

Findings

Some rigid surfaces have no real structures.

Existence of a rigid surface with a non-maximal real structure.

No real surfaces among surfaces of general type with $p_g=q=0$ and $K^2=9$.

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.