Fourier decay, Green's kernel and Schottky groups

TL;DR

This paper explores a method for constructing Green's functions on algebraic surfaces via Schottky uniformization, analyzing convergence and geometric interpretation to improve understanding of the approach.

Contribution

It provides a detailed investigation into the convergence of deformations of Green's function formulas using Schottky groups, clarifying their geometric meaning.

Findings

Convergence of deformed Green's function formulas is established.

Geometric interpretation of Schottky uniformization in this context is clarified.

Potential applications to algebraic surface analysis are discussed.

Abstract

In this note, I would like to discuss an approach to the construction of Green's function on algebraic surfaces, indicated by Manin, towards the computation of the Green's function on surfaces using Schottky uniformization. We shall see that the exact geometric interpretation of the formula mentioned there is obscure, and try to remedy the situation by investigating convergence of deformations of that formula.