Vector fields and foliations on compact surfaces of class \(\text{VII}_0\) (Q1805925)

Let GSS be the class of compact complex surfaces \(S\) containing global spherical shells. If \(S\) is GSS then it contains exactly \(n=b_2(S)\) rational curves \(D_0,\ldots,D_{n-1}\) each of them being regular or with a double point. Let \(\sigma(S):=-\sum_{i=0}^{n-1}D_i^2+2\text{Card}\{\text{double points}\}\). Then \(2n\leq\sigma(S)\leq 3n\). The main result proved by the authors is the following: let \(S\) be GSS and minimal; then there is always a global singular holomorphic foliation on \(S\). Furthermore if \(b_2(S)\geq 1\) then \(S\) admits at most two foliations and there are two foliations iff \(S\) is an Inoue-Hirzebruch surface. If \(2n<\sigma(S)<3n\) and there exists a numerically anticanonical divisor, there exists a logarithmic deformation of \(S\) into a surface admitting a global non-trivial vector field.