Some General Properties of Surfaces

In the fundamental Sect. 1 we treat meromorphic maps induced by line bundles. Then, in Sect. 2, we deal with special features for differential forms on compact surfaces. The main point is that for surfaces the Fröhlicher spectral sequence always degenerates. Combining the consequences of this fact with the topological index theorem we find, following Kodaira, relations between topological and analytical invariants which are crucial in handling non-algebraic surfaces. For instance, in Sect. 3 we give a direct proof (due to Lamari) that a compact surface is Kähler if and only if its first Betti number is even.