Real structures on primary Hopf surfaces

Abstract

Abstract The first goal of this article is to give a complete classification (up to Real biholomorphisms) of Real primary Hopf surfaces ( H , s ) \left(H,s) , and, for any such pair, to describe in detail the following naturally associated objects : the group Aut h ( H , s ) {{\rm{Aut}}}_{h}\left(H,s) of Real automorphisms, the Real Picard group ( Pic ( H ) , s ˆ * ) \left({\rm{Pic}}\left(H),{\hat{s}}^{* }) , and the Picard group of Real holomorphic line bundles Pic R ( H ) {{\rm{Pic}}}_{{\mathbb{R}}}\left(H) . Our second goal is the classification of Real primary Hopf surfaces up to equivariant diffeomorphisms, which will allow us to describe explicitly in each case the real locus H ( R ) = H s H\left({\mathbb{R}})={H}^{s} and the quotient H ⁄ ⟨ s ⟩ H/\langle s\rangle .