Index of Kato surfaces

The compact curves of an intermediate Kato surface $S$ form a basis of $H^2(S,\mathbb Q)$. We present a way to compute the associated rational coefficients of the first Chern class $c_1(S)$. We get in particular a simple geometric obstruction for $c_1(S)$ to be an integral class, or equivalently index$(S)=1$. We also find an expression for the exponents of the contracting germ of $S$ in terms of self-intersection numbers of the compact curves.