Periods in Families and Derivatives of Period Maps

TL;DR

This paper investigates the algebraic and transcendental properties of periods and their derivatives in families of algebraic varieties, revealing their relation to Hodge structures and generalizing previous work by Bertrand--Zudilin.

Contribution

It establishes a connection between periods, their derivatives, and Hodge filtration coordinates, extending the understanding of their algebraic independence and transcendence in families.

Findings

The field generated by periods and their derivatives matches that of Hodge filtration coordinates.

Determines the transcendence degree of flat coordinate fields for algebraic sections in flat vector bundles.

Generalizes previous results by Bertrand--Zudilin to broader contexts.

Abstract

Given a smooth proper family $\phi:X\rightarrow S$, we study the (quasi)-periods of the fibers of $\phi$ as (germs of) functions on $S$. We show that they field they generate has the same algebraic closure as that given by the flag variety co-ordinates 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.