From the author’s abstract: Here we will concentrate on a specialized aspect of the theory of compact Riemann surfaces – namely the construction and properties of ‘Green’s functions’. These functions (on Riemann surfaces as well as on abelian varieties) turn out to be crucial for the development of arithmetic geometry through their connection with the so-called “Néron families” and “Weil functions” – which are close relatives of Green’s. Our main aim is to obtain the construction of these functions from various techniques – and exhibit the ‘interrelating connections’. One main consequence, of independent interest, that we will derive below is the existence of a canonical hermitian metric (canonical up to scalars) on each holomorphic line bundle over a compact Riemann surface.