## Burniat surfaces. III: Deformations of automorphisms and extended Burniat surfaces

### Summary

Summary: We continue our investigation of the connected components of the moduli space of surfaces of general type containing the Burniat surfaces, correcting a mistake in part II. We define the family of extended Burniat surfaces with $K_S^2 = 4$, resp. 3, and prove that they are a deformation of the family of nodal Burniat surfaces with $K_S^2 = 4$, resp. 3. We show that the extended Burniat surfaces together with the nodal Burniat surfaces with $K_S^2=4$ form a connected component of the moduli space. We prove that the extended Burniat surfaces together with the nodal Burniat surfaces with $K_S^2=3$ form an irreducible open set in the moduli space. Finally we point out an interesting pathology of the moduli space of surfaces of general type given together with a group of automorphisms $G$. In fact, we show that for the minimal model $S$ of a nodal Burniat surface $(G = (\ZZ/2 \ZZ)^2)$ we have $\Def(S,G) \neq \Def(S)$, whereas for the canonical model $X$ it holds $\Def(X,G) = \Def(X)$. All deformations of $S$ have a $G$-action, but there are different deformation types for the pairs $(S,G)$ of the minimal models $S$ together with the $G$-action, while the pairs $(X,G)$ have a unique deformation type.

### Mathematics Subject Classification

14J29, 14J25, 14J10, 14D22, 14H30, 32G05