Functional Transcendence of Periods and the Geometric Andr\'e--Grothendieck Period Conjecture

TL;DR

This paper establishes a functional transcendence theorem for algebraic integrals in families of varieties, leading to a geometric proof of a generalized Grothendieck period conjecture using Nori motives.

Contribution

It proves a new functional transcendence theorem and a geometric version of Andre9's period conjecture, extending Ax--Schanuel results to mixed Hodge structures.

Findings

Proved a functional transcendence theorem for algebraic integrals.

Established a geometric version of Andre9's period conjecture.

Extended Ax--Schanuel theorems to mixed Hodge structures.

Abstract

We prove a functional transcendence theorem for the integrals of algebraic forms in families of algebraic varieties. This allows us to prove a geometric version of Andr\'e's generalization of the Grothendieck period conjecture, which we state using the formalism of Nori motives. More precisely, we prove a version of the Ax--Schanuel conjecture for the comparison between the flat and algebraic coordinates of an arbitrary admissible graded polarizable variation of integral mixed Hodge structures. This can be seen as a generalization of the recent Ax--Schanuel theorems of \cite{chiu,GaoKlingler} for mixed period maps.