On surfaces of class VII+0 with numerically anticanonical divisor

We consider minimal compact complex surfaces S with Betti numbers b₁ = 1 and n = b₂ > 0. A theorem of Donaldson gives n exceptional line bundles. We prove that if in a deformation, these line bundles have sections, S is a degeneration of blown-up Hopf surfaces. Besides, if there exists an integer m ≥ 1 and a flat line bundle F such that H⁰(S,-mK ⊗ F) ≠ 0, then S contains a Global Spherical Shell. We apply this last result to complete classification of bihermitian surfaces.