PERIODS IN FAMILIES AND DERIVATIVES OF PERIOD MAPS

Abstract

Given a smooth proper family

, we study the (quasi)-periods of the fibres of f as (germs of) functions on S . We show that the field they generate has the same algebraic closure as that given by the flag variety coordinates parametrizing the corresponding Hodge filtration, together with their derivatives. Moreover, in the more general context of an arbitrary flat vector bundle, we determine the transcendence degree of the function field generated by the flat coordinates of algebraic sections. Our results are inspired by and generalize work of Bertrand–Zudilin.