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).
Fujiki, A., Pontecorvo, M. (2015). Numerically anticanonical divisors on Kato surfaces. J. Geom. Phys. 91 (2015), 117–130. JOURNAL OF GEOMETRY AND PHYSICS, 91, 117-130 [10.1016/j.geomphys.2015.01.001].