On Hartshorne's problem for compact C-analytic surfaces M with κ(M)=−∞

AbstractLet M be a compact C-analytic surface, let Γ⊂M be a compact analytic subvariety and let X:=M\Γ. The following two problems will be considered: Assume that X does not contain any compact curve and that Γ is an irreducible compact curve in X such that Γ2≥0 (respectively assume that the analytic cohomology groups H1(X,Ωp)=0, for all 0≤p≤2). Is X always Stein? It is our main purpose here to provide an affirmative answer to those two problems, provided M is a (minimal) compact C-analytic surface such that its Kodaira dimension κ(M)=−∞. Also the affine structure of such Stein surfaces will be discussed