SURFACES WITH K 2 = 8, pg = 4 AND CANONICAL INVOLUTION.

The aim of this paper is to classify regular minimal surfaces S with K = 8 and pg = 4 whose canonical map factors through an involution (short: having a canonical involution). The study of surfaces with geometric genus pg = h (S,ΩS) = 4 began with Enriques’ celebrated book Le superficie algebriche ([Enr]), where he summarized his research of over fifty years. By standard inequalities, minimal surfaces with geometric genus pg = 4 satisfy 4 ≤ K 2 S ≤ 45. While for high values of K 2 S it is already difficult to prove existence, the challenge for low values is to completely classify all surfaces with the given value of K S. More ambitiously, one would like to understand the topology of the moduli space, i.e., the irreducible and connected components of the moduli space. The lowest possible values K S = 4, 5 were already treated by Enriques and the corresponding moduli spaces were completely understood in the 70’s. For K S = 6 the situation is far more complicated. In [Hor3] Horikawa completely classifies all surfaces with pg = 4 and K = 6, obtaining a stratification of the moduli space in 11 strata. Moreover he shows that there are 4 irreducible components, and at most three connected components. In [BCP] it is shown that the number of connected components actually cannot be bigger than two. Let us point out that all these surfaces are homeomorphic. The complete classification of minimal surfaces with K S = 7 and pg = 4 was achieved by the first author in [Bau]. Moreover, it is shown there that all these surfaces are homeomorphic, and that there are three irreducible components and at most two connected components.