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 $\tau\in H^0(S,\Omega^1(\log D)\otimes \mathcal L_\lambda)$, 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 $\hat \tau$ on the cyclic covering $p:\hat S\to S$ of $S$ which is PH outside $\hat D:=p^{-1}(D)$ with automorphy constant being the same automorphy constant $\lambda$ for a suitable automorphism of $\hat S$. With previous results obtained with V.Apostolov it suggests a strategy to prove the GSS conjecture.