On a theorem of Faltings on formal functions

TL;DR

This paper revisits Faltings' 1980 theorem on formal functions, providing two elementary geometric proofs of a key extension property for rational functions along subvarieties.

Contribution

The paper offers two simplified, elementary proofs of Faltings' theorem on formal functions, making the result more accessible.

Findings

Formal rational functions on certain subvarieties extend uniquely to the entire variety.

Elementary geometric proofs are feasible for Faltings' theorem.

The results hold over algebraically closed fields of any characteristic.

Abstract

In 1980, Faltings proved, by deep local algebra methods, a local result regarding formal functions which has the following global geometric fact as a consequence. Theorem: Let k be an algebraically closed field (of any characteristic). Let Y be a closed subvariety of a projective irreducible variety X defined over k. Assume that X \subseteq P^n, dim(X)=d>2 and Y is the intersection of X with r hyperplanes of P^n, with r \le d-1. Then, every formal rational function on X along Y can be (uniquely) extended to a rational function on X. Due to its importance, the aim of this paper is to provide two elementary global geometric proofs of this theorem.