Definability of mixed period maps

Abstract

We equip integral graded-polarized mixed period spaces with a natural ${\mathbb{R}}_{\mathrm{alg}}$-definable analytic structure, and prove that any period map associated to an admissible variation of integral graded-polarized mixed Hodge structures is definable in ${\mathbb{R}}_{\mathrm{an},\exp }$ with respect to this structure. As a consequence we re-prove that the zero loci of admissible normal functions are algebraic.