On Classification of compact complex surfaces of class VII

Let $S$ be a minimal compact complex surface with Betti numbers $b_1(S)=1$ and $b_2(S)\ge 1$ i.e. a compact surface in class VII${}_0^{+}$. We show that if there exists a twisted logarithmic 1-form $\tau \in H^0(S,{\mathrm{\Omega }}^1(\log D)\otimes {\mathcal{L}}_{\lambda })$, where $D$ is a non zero divisor and $\mathcal{L}\in H^1(S,{\mathbb{C}}^{\star })$, then $S$ is a Kato surface. It is known that $\lambda$ is in fact real and we show that $\lambda \ge 1$ and unique if $S$ is not a Inoue-Hirzebruch surface. Moreover $\lambda =1$ if and only if $S$ is a Enoki surface. When $\lambda >1$ these conditions are equivalent to the existence of a negative PSH function $\widehat{\tau }$ on the cyclic covering $p:\widehat{S}\to S$ of $S$ which is PH outside $\widehat{D}:=p^{-1}(D)$ with automorphy constant being the same automorphy constant $\lambda$ for a suitable automorphism of $\widehat{S}$. With previous results obtained with V.Apostolov it suggests a strategy to prove the GSS conjecture.