On Classification of Compact Complex Surfaces of Class VII

Let S be a minimal compact complex surface with Betti numbers b₁(S)=1 and b₂(S)≥ 1 i.e. a compact surface in class VII₀⁺. We show that if there exists a twisted logarithmic 1-form τ∈ H⁰(S,Ω¹(log D)⊗ L_λ), where D is a non zero divisor and L∈ H¹(S, C⁾, then S is a Kato surface. It is known that λ is in fact real and we show that λ≥ 1 and unique if S is not a Inoue-Hirzebruch surface. Moreover λ=1 if and only if S is a Enoki surface. When λ>1 these conditions are equivalent to the existence of a negative PSH function τ on the cyclic covering p: S→ S of S which is PH outside D:=p⁻¹(D) with automorphy constant being the same automorphy constant λ for a suitable automorphism of S. With previous results obtained with V.Apostolov it suggests a strategy to prove the GSS conjecture.