Numerically anticanonical divisors on Kato surfaces

The compact curves of an intermediate Kato surface S form a basis of H2(S,Q). We present a way to compute the associated rational coefficients of the first Chern class c1(S). We get in particular a simple geometric obstruction for c1(S) to be an integral class, or equivalently index (S)=1. In the final part we discuss relations with some recent work of Dloussky (2011) and Oeljeklaus and Toma (2009).