Periods of E-operators

Abstract

We define an enlargement of the class of exponential periods by applying the rapid decay cohomology theory of Hien to a larger class of integrable connections on varieties than those twisted by a regular function. The main and motivating examples are given by those associated to $E$-operators. This delivers a method to realise a class of absolutely convergent integrals involving $E$-functions as matrix elements of a period pairing, which we call \emph{$E$-periods}. Furthermore, we generalise rapid decay cohomology to singular varieties and prove a version of Nori's basic lemma in this setting.