Green’s Functions on Riemann Surfaces

In this chapter, we give some explicit formulas for Néron functions on Riemann surfaces, and an explicit construction due to Coleman. These are with respect to a canonical volume form. We shall also explain how the functions change when we change the metrics. We work complex analytically, in which case we can characterize the Néron functions in a complex analytic fashion, thus finding classical objects called Green’s functions. We shall give the definition of Green’s function and prove its basic properties ab ovo. Actually, we give several proofs for some of the basic theorems, depending on different explicit constructions. One of them depends on the Hodge decomposition and harmonic forms, for which a complete treatment is given in Griffiths-Harris. Another depends on the smoothness of the Green’s function, which may be constructed by theta functions. Different people at different times will use the different techniques for different purposes. Since the construction of the Green’s function by theta functions is given in detail in [La 1], Chapter 13, I do not reproduce this construction here. For the most part, in the application to the intersection theory and Riemann-Roch theorem, we use only the basic formal properties, and the construction of a Green’s function is irrelevant. In the proof of the existence of Faltings volumes, given in Chapter VI, we need to relate the Green’s function on the curve with the Green’s function on the Jacobian, associated with the theta divisor. At this point, we shall make a reference to [La 1], Chapter 13 for one particular property that is needed.