Generalized Burniat type surfaces and Bagnera-de Franchis varieties (Q2788586)

This paper is devoted to the construction and classification of certain minimal surfaces of general type with \(\chi({\mathcal O})=1\), the minimal possible value.NEWLINENEWLINEFirst the authors study a variant of a construction due to Inoue, who contructed certain surfaces (already obtained by Burniat as covers of the plane) as quotient by a finite group of invariant hypersurfaces in a product of three elliptic curves \(\prod_1^3 E_i\). Writing the three curves as complete intersection of two quadrics, both ot the form \(\sum a_i x_i^2=0\), one easily sees an automorphism group of \(\prod_1^3 E_i\) isomorphic to \(\left( {\mathbb Z}/2{\mathbb Z} \right)^9\).NEWLINENEWLINEThe paper considers a subgroup \({\mathcal H}' \cong \left( {\mathbb Z}/2{\mathbb Z} \right)^3\) of it such that \(\left( \prod_1^3 E_i \right)/{\mathcal H}'\cong \left( {\mathbb P^1}\right)^3\); it inherits naturally an action of a group \({\mathcal H} \cong \left( {\mathbb Z}/2{\mathbb Z} \right)^6\). Then the authors classify subgroups of \({\mathcal H}\) or order \(4\) or \(8\) leaving an irreducible Del Pezzo surface in \(|{\mathcal O}_{\left( {\mathbb P^1}\right)^3}(1,1,1)|\) invariant. Next they consider the preimage of this Del Pezzo surface in \(\prod_1^3 E_i\), which is then invariant by a group of the form \(\left( {\mathbb Z}/2{\mathbb Z}\right)^a\) for \(a=5\) or \(6\), and classify for which groups the action is free.NEWLINENEWLINEThey denote finally by \textit{generalized Burniat surfaces} the surfaces obtained as quotient by such a free action and classifiy them, finding \(16\) irreducible families of minimal surfaces of general type with \(\chi({\mathcal O})=1\), \(K^2=6\).NEWLINENEWLINEFor all these families they compute \(p_g=q\) and the fundamental group, deducing from it that most of these families realize surfaces of general type of a topological type that was not known before.NEWLINENEWLINEThey also compute the small deformations of a general surface in each family, deducing in most cases that their family dominates a component of the Gieseker moduli space of the canonical models of surfaces of general type.NEWLINENEWLINEThe only new cases in which their family does not dominate a component of that moduli space are two cases that led them to the definition of \textit{Sicilian surfaces}, minimal surfaces of general type with \(p_g=q=1\), \(K^2=6\) having a birational étale bidouble cover with \(q=3\) and birational Albanese morphism. They classify these surfaces, obtaining a component of dimension \(4\) of the Gieseker moduli space, containing two families of generalized Burniat surfaces as irreducible divisors. They also show that each surface homotopically equivalent to a Sicilian surface is a Sicilian surface.