On a theorem of Faltings on formal functions
  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. 
Paola Bonacini, Alessio del Padrone, Michele Nesci