On Classification of compact complex surfaces of class VII

Abstract

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 $\ddot{I}„\in H^0(S,\hat{I}{\text{c◯}}^1(\log D)\otimes {\mathcal{L}}_{\hat{I}}»)$, 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 $\hat{I}»$ is in fact real and we show that $\hat{I}»\ge 1$ and unique if $S$ is not a Inoue-Hirzebruch surface. Moreover $\hat{I}»=1$ if and only if $S$ is a Enoki surface. When $\hat{I}»>1$ these conditions are equivalent to the existence of a negative PSH function $\hat{\ddot{I}}„$ 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 $\hat{I}»$ for a suitable automorphism of $\hat{S}$. With previous results obtained with V.Apostolov it suggests a strategy to prove the GSS conjecture.

24 pages, proof of Thm 4.2 is more detailed