Class VII surfaces with $b_2$ curves
  We give an affirmative answer to a conjecture of Ma. Kato, namely that every compact complex surface $S$ in Kodaira's class $VII_0$ with $b_2(S) > 0$ and $b_2(S)$ rational curves, admits a global spherical shell. 
Georges Dloussky, Karl Oeljeklaus and Matei Toma