Geometric Differential Equations

Abstract

In this chapter we explain in a precise way what should be the differential equations satisfied by G-functions according to the conjecture stated in the introduction : namely, the “geometric” differential equations over Φ̄ . These are combinations of factors of Picard-Fuchs equations attached to proper smooth varieties defined over Φ̄(x) .We study the stability of this class of differential equations under standard operations and show that their solutions in Φ̄[[x]] form a Φ̄-vector space stable under Cauchy and Hadamard products (making use of Hodge theory). We shall show in chapter 5 that such solutions are indeed G-functions.