Ax-Schanuel for variations of mixed Hodge structures

TL;DR

This paper establishes the Ax-Schanuel property for period mappings from variations of mixed Hodge structures, utilizing definability and rational point counting on definable quotients.

Contribution

It proves the Ax-Schanuel property for mixed Hodge structures' period mappings, extending previous results with new definability and point counting techniques.

Findings

Proves Ax-Schanuel property for mixed Hodge structures

Uses definability of mixed period mappings in the proof

Counts rational points on definable quotients

Abstract

We give properties of the real-split retraction of the mixed weak Mumford-Tate domain and prove the Ax-Schanuel property of period mappings arising from variations of mixed Hodge structures. An ingredient in the proof is the definability of the mixed period mapping obtained by Bakker-Brunebarbe-Klingler-Tsimerman. In comparison with preceding results, in the point counting step, we count rational points on definable quotients instead.